diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzqino" "b/data_all_eng_slimpj/shuffled/split2/finalzzqino" new file mode 100644--- /dev/null +++ "b/data_all_eng_slimpj/shuffled/split2/finalzzqino" @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\nAn inevitable consequence of the N-body dynamics near a supermassive black hole (BH) is that some stars will be scattered into highly eccentric orbits and get disrupted by the BH's enormous tidal forces \\citep{hills75_tdes_agn, frank76_tde_rate, magorrian99_tde_rate, alexander05_stellar_dynamics, stone20_tde_rate_review}. Such tidal disruption events (TDEs) feed a transient episode of accretion onto the otherwise dormant BH and generate a bright flare across the electromagnetic spectrum \\citep{lacy82_tdes_Galactic_Center, rees88_tdes, phinney89_tde_fallback_rate,cannizzo90_tde_accretion, ulmer99_tde_flare, strubbe09_tde_flare}. They provide our best chance of probing the demographics of supermassive BHs at the centers of quiescent galaxies at cosmological distances \\citep{kormendy13_SMBH_demographics}.\n\nThe hydrodynamics of the tidal disruption process has now been reasonably well understood, thanks to a large volume of analytical and numerical works \\citep{carter83_tidal_compression, evans89_tde_simulation, ayal00_tde_simulation, lodato09_tde_simulation, guillochon13_tde_simulation, stone13_tde_fb_rate, cheng14_relativistic_tdes, tejeda17_relativistic_tdes, steinberg19_deep_tdes, ryu20_tde_in_GR, rossi20_disruption_process}. The consequence is that the star is tidally stretched into a long, thin stream of debris, the evolution of which is controlled by the tidal gravity, self-gravity, as well as internal pressure forces \\citep{kochanek94_stream_evolution,coughlin16_stream_evolution, Bonnerot21_stream_width_evolution}. In the longitudinal direction of the stream, the evolution is almost entirely controlled by tidal gravity, meaning that each segment of the stream accurately follows its own geodesic like a test particle. Each time a fluid element passes near the pericenter, it undergoes general relativistic (GR) precession, which will eventually lead to the self-intersection of the stream \\citep{rees88_tdes, dai13_gr_precession, dai15_pn1_precession, guillochon15_dark_year, Bonnerot17_stream_crossings, liptai19_GRSPH_simulation, lu20_self-intersection}. If this intersection is sufficiently violent, it strongly broadens the angular momentum and orbital energy distributions of the bound debris and leads to numerous secondary shocks and then the formation of a coherently rotating accretion flow \\citep{shiokawa15_GR_precession_intersection, bonnerot16_precession_schwarzschild, hayasaki16_precession_kerr, sadowski16_relativistic_tde, jiang16_stream_collision, lu20_self-intersection, bonnerot20_disk_formation, andalman20_disk-formation, curd21_global_TDE, bonnerot21_accretion_flow_review}. The processes of disk formation and accretion generate bright electromagnetic emission that is observable from cosmological distances \\citep{strubbe09_tde_flare, coughlin14_ZEBRA, piran15_stream_collision, metzger16_bright_year, dai18_unified_model, lu20_self-intersection, bonnerot21_first_light}.\n\nOn the observational side, of the order $10^2$ TDE candidates have been identified by various transient searching strategies in different bands, including X-ray surveys \\citep{bade96_rosat_tde, esquej08_xmm_newton_tdes, saxton12_x-ray_tde, saxton17_xmm_tde, lin15_xmm_tde, brightman21_X-ray_TDE, sazonov21_erosita_TDEs}, UV-optical surveys \\citep{gezari06_GALEX_tde, gezari12_ps-10jh, chornock14_ps1-11af, holoien16_as14li, blagorodnova17_iptf16fnl, hung17_iPTF16axa, hinkle21_AS19dj, vanvelzen21_ztf_tdes}, and infrared surveys \\citep{wang18_infrared_tdes, jiang21_MIRONG}.\nTo extract the physical information from the increasing number and diversity of TDEs, a better understanding of the overall hydrodynamics is needed. A crucial ingredient is the self-intersection of the fallback stream, as it initiates the formation of the accretion disk and all subsequent electromagnetic emission. \n\nIn this work, we compute the geodesic motion of the fallback stream in the Kerr spacetime for a broad range of BH spins, orbital angular momenta and inclinations. An important feature of an inclined Kerr geodesic is that Lense-Thirring (LT) precession may cause missed prompt collision right after the first pericenter passage \\citep{dai13_gr_precession, guillochon15_dark_year, hayasaki16_precession_kerr}, unlike in the Schwarzschild case. While the motion of the stream center is described by a Kerr geodesic, we model the stream thickness by integrating an approximated tidal equation. Then, the position of the stream collision is where the separation between the stream centers is less than the sum of the thicknesses of the two approaching ends. One of the key differences between our model and the work of \\citet{guillochon15_dark_year} is the treatment of stream thickness evolution. They assume that the stream thickness increases linearly with orbital winding number (their Eq. 2), which they attribute to some unspecified energy dissipation by the nozzle shock near the pericenter. Consequently, the stream collision is mainly the result of the continuing growth of the stream thickness in their study. However, recent numerical simulations by \\citet{Bonnerot21_stream_width_evolution} show that the stream thickness does not dramatically increase after it passes through the nozzle shock. In this work, we treat the effect of the nozzle shock as a perfect bounce, and the evolution of the stream thickness is determined by the relativistic tidal forces.\nIt will be shown that the physical reason for the stream collision is a geometrical one that can be analytically described in a post-Newtonian picture for mildly relativistic orbits.\n\nThis paper is organized as follows. In \\S \\ref{sec:methods}, we describe our algorithm, which takes as input the BH spin and the stream's orbital parameters and then computes the position of the stream intersection. A parameter space exploration is presented in \\S \\ref{sec:parameter_exploration}, where we show how the properties of the stream intersection depend on the BH spin and the stream's orbital angular momentum and inclination. In \\S\\ref{sec:physical_reason}, we provide a post-Newtonian description that unveils the physical reason for the stream collision and analytically predicts the delay time before the collision. The limitations of our model are discussed in \\S \\ref{sec:discussion}. A summary of the main results is provided in \\S \\ref{sec:summary}. In this work, we only consider a representative TDE case of a $M=10^6M_\\odot$ BH and a $1M_\\odot$ star. The exploration of a wider parameter space is left for a future work. Our calculations are fully general relativistic, so we use geometrical units with $G=M=c=1$ throughout the paper, unless otherwise specified.\n\n\n\\section{Methods}\\label{sec:methods}\n\n\\subsection{Geodesic motion of stream center}\n\\label{sec:orbit}\n\nThe geodesic of a representative debris particle at the center of the fallback stream is specified by the following parameters:\n\\begin{enumerate}\n \\item $\\tilde E$, the total specific energy (including rest mass),\n \\item $L$, the total specific angular momentum,\n \\item $I$, the inclination of the orbit (where $I=0$ is for the equatorial case and $I=\\pi\/2$ is for an initially polar orbit), and\n \\item $a$, the BH spin (where $a=0$ is the Schwarzschild case and $a<0$ is for retrograde orbits),\n\\end{enumerate}\nThe inclination angle is defined by $\\cos I = L_z\/L$ ($L_z$ being the specific angular momentum component along the spin axis) and the Carter constant is $Q=L^2(\\sin I)^2$.\n\nThe geodesic equations of motion for a Kerr spacetime are numerically integrated to machine precision using a program based on \\cite{1994ApJ...421...46R}. We work in Boyer-Lindquist coordinates ($t$, $r$, $\\theta$, $\\phi$) with the metric given by\n\\begin{eqnarray}\n\\begin{split}\n d s^2 = \\,& -\\left(1-{2r\\over \\Sigma}\\right) dt^2 - {4ar\\sin^2\\theta \\over \\Sigma} dt d\\phi + {\\Sigma \\over \\Delta} dr^2 + \\Sigma d \\theta^2 \\\\\n &+ \\left(r^2 + a^2 + {2a^2 r\\sin^2\\theta \\over \\Sigma}\\right)\\sin^2\\theta\\, d \\phi^2,\n\\end{split}\n\\end{eqnarray}\nwhere\n\\begin{eqnarray}\n \\Sigma=r^2+a^2\\mu^2,\n \\label{eq:sigma}\n\\end{eqnarray}\n\\begin{eqnarray}\n \\Delta=r^2-2r+a^2.\n \\label{eq:delta}\n\\end{eqnarray}\nThe orbit is initialized at the apocenter at $\\tau=\\phi=0$ and $\\theta=\\pi\/2$, where $\\tau$ is the proper time. Note that starting from $\\theta=\\pi\/2$ (in the BH's equatorial plane) loses the generality of the initial conditions, but as we show in \\S\\ref{sec:physical_reason}, this choice is representative for most TDEs. The main effect of other choices of initial $\\theta$ is to change the delay time before the stream collision, and the dependence can be analytically predicted under the post-Newtonian approximation (see \\S\\ref{sec:physical_reason}).\n\nFor $u=1\/r$ and $\\mu=\\cos\\theta$, points on the orbit are sampled in intervals of $du$, which is related to proper-time intervals $d\\tau$ by \\citep{bardeen72_Kerr_geodesic}\n\\begin{eqnarray}\n \\label{eq:dudtau}\n \\frac{du}{d\\tau}=u^2\\frac{\\sqrt{V_r}}{\\Sigma},\n\\end{eqnarray}\nwhere\n\\begin{eqnarray}\n \\label{eq:Vr}\n V_r=T^2-\\Delta \\left(r^2+K\\right),\n\\end{eqnarray}\n\\begin{eqnarray}\n K = (L_z - a\\tilde{E})^2 + Q,\n\\end{eqnarray}\n\\begin{eqnarray}\\label{eq:T}\n T=\\tilde E \\left(r^2+a^2\\right)-aL_z.\n\\end{eqnarray}\nAn example of the geodesic motion for total energy $\\tilde E=0.9999$, total angular momentum $L=6.5$, inclination angle $\\cos I=0.5$ and BH spin $a=0.9$ is shown in Fig. \\ref{fig:fullorbit}.\n\\begin{figure}\n \\centering\n \\includegraphics[width=0.48\\textwidth]{mode1_full.png}\n \\caption{The orbit around a Kerr BH for a test particle at the center of the fallback stream. The parameters for this case are: total energy $\\tilde E=0.9999$, total angular momentum $L=6.5$, inclination angle $\\cos I=0.5$, BH spin $a=0.9$. The position of the BH, the origin of the Cartesian coordinate system, is marked by a black dot. The evolution is terminated when the stream crossing occurs, as marked by a red dot (the parameters for the stream thickness evolution are described in \\S \\ref{sec:stream_thickness}). All axes are in units of the gravitational radius $GM\/c^2$. \n } \n \\label{fig:fullorbit}\n\\end{figure}\n\n\\subsection{Stream thickness evolution}\\label{sec:stream_thickness}\n\n\\begin{figure}\n \\centering\n \\includegraphics[width=0.48\\textwidth]{r1.pdf}\n \\includegraphics[width=0.48\\textwidth]{r2.pdf}\n \\includegraphics[width=0.48\\textwidth]{r3.pdf}\n \\caption{The evolution of stream thickness $H$ as a function of $r$, magnified further to show details. To read the plot, start at the apocenter, and follow the plot continuously to see how the thickness would evolve in time and with $r$. On average, the stream becomes thicker with each successive winding, and the thickness ratio between the outward and inward segments of the same winding decreases with time. The geodesic is the same as shown in Fig. \\ref{fig:fullorbit}, with $\\tilde E=0.9999$, $L=6.5$, $\\cos I=0.5$, $a=0.9$. Here we set $\\sigma=1$ (see \\S \\ref{sec:stream_thickness} for its definition). The evolution is terminated when the stream crossing occurs, as marked by the red dots. Both axes are in units of the gravitational radius $GM\/c^2$. \n }\n \\label{fig:H_evolve}\n\\end{figure}\n\nAfter tidal disruption, the star is stretched into a long, thin stream. In the previous subsection, we compute the orbit of a test particle at the center of mass of the stream. In the plane transverse to the velocity vector, the stream has finite thickness, the evolution of which must be calculated to obtain the position of stream crossing. The long-term evolution of the stream thickness over multiple pericenter passages is currently still an open question, due to the prohibitive computational costs of global GR (magneto-) hydrodynamic simulations \\citep[see][and references therein]{bonnerot21_accretion_flow_review}. Right after the tidal disruption, self-gravity plays an important role confining the stream in the transverse direction, but it becomes unimportant soon after the first apocenter passage. We therefore neglect the influence of pressure forces in the subsequent evolution except at the nozzle shocks caused by short episodes of strong compression in the transverse direction.\n\nThese considerations motivate us to model the evolution of the stream thickness $H$ according to the following approximate tidal force equation based on the $C_{22}$ component of the tidal tensor given by \\citep[][their Eq. (56)]{marck83_tidal_tensor}\n\\begin{eqnarray}\\label{eq:tfe}\n\\begin{split}\n \\frac{d^2H}{d\\tau^2} &\\approx -H {r\\over \\Sigma^3}(r^2 - 3a^2\\mu^2)\n \\left(1 + 3{r^2 R^2 - a^2\\mu^2 S^2 \\over K \\Sigma^2}\\right),\\\\\n R &= K - a^2\\mu^2,\\ S = r^2 + K,\n\\end{split}\n\\end{eqnarray}\nwhere the stream thickness $H$ is considered to be in the direction perpendicular to the local orbital plane (as defined far from the BH where GR is unimportant). Eq. (\\ref{eq:tfe}) is exact in the Schwarzschild limit ($a\\approx 0$), which is a good approximation because most of the stream-thickness evolution occurs far from the BH where the spin effects are minor. For the majority of the parameter space considered in this work, the LT precession angle is small ($\\ll 1\\rm\\, rad$) and stream crossing typically occurs between adjacent orbital windings, so the thickness perpendicular to the local orbital plane is the main physical quantity determining the occurrence of intersection. We do not consider the evolution of stream thickness within the local orbital plane, which is affected by the angular momentum spread across the stream\\footnote{The angular momentum spread across the stream \\citep[as can be seen from the simulations by][]{cheng14_relativistic_tdes} may cause the stream to look like a fan, with in-plane width typically larger than the vertical width. } as well as more complicated tidal and non-inertial forces than in Eq. (\\ref{eq:tfe}). A full treatment of the 2D stream cross-sectional structure in Kerr spacetime is much more complicated and computationally expensive, and this is left for future works.\n\n\n\nWe consider the part of the stream that is initially self-gravitating right after the disruption.\nThe initial conditions for the thickness evolution are taken to roughly correspond to the physical situation that the stream's self-gravity becomes sub-dominant compared to tidal forces shortly after the first apocenter passage. Motivated by the hydrodynamic simulations of \\citet[][their Fig. 5]{Bonnerot21_stream_width_evolution}, we set the initial stream thickness $H_0=5R_\\odot$ and the initial vertical expansion velocity $\\dot H_0=0$ when the stream center reaches a distance $r=r_{\\rm a}\/2$ ($r_{\\rm a}$ being the apocenter radius) from the BH after the first apocenter passage. Our results are insensitive to the exact position of maximum thickness (where $\\dot H=0$), because the thickness only depends weakly on the radius for this part of the evolution.\nFor the same reason, we also take the stream thickness between $r=r_{\\rm a}\/2$ and $r=r_{\\rm a}$ (the apocenter) to be constant at $H=H_0$. As we will discuss later, the line tangent to the orbit at the point of maximum stream thickness (where $\\dot H = 0$) determines the positions of the nozzle shocks (where $H=0$). Since we choose the initial point for the thickness evolution to be on the semi-\\textit{minor} axis, this tangent line is parallel to the semi-major axis. For other choices of the initial point somewhere between $r=r_{\\rm a}\/2$ and $r_{\\rm a}$ (but not extremely close to the apocenter), the tangent line will also be approximately parallel to the major axis, because the orbit is nearly radial due to its high eccentricity. In the subsequent evolution, the positions of the nozzle shocks are strongly affected by apsidal precession and are insensitive to our choice of the initial point.\n\n\nFor a given geodesic with known $r(\\tau)$ and $\\theta(\\tau)$, we numerically integrate Eq. (\\ref{eq:tfe}) to find the stream thickness evolution $H(\\tau)$. The integration is performed using the fourth-order Runge-Kutta method with an adaptive step size of $d\\tau=10^{-4}r^{3\/2}$ (and accuracy is checked through a numerical convergence study).\nWe then map $d\\tau$ to $du$ using Eq. (\\ref{eq:dudtau}).\n\n\nAnother aspect of the stream thickness evolution is the occurrence of nozzle shocks when the thickness reaches $H=0$. Near the point of maximum compression, hydrodynamic shocks convert the bulk kinetic energy associated with the motion perpendicular to the local orbital plane into heat. The pressure of the shocked gas then leads to outward expansion. As long as the timescale for the expansion of the shocked gas is much shorter than the orbital time near the pericenter, the effect of the nozzle shocks may be approximately taken into account by instantaneously flipping the sign of $\\dot H$. Our model assumes the post-bounce vertical velocity to be $-\\sigma \\dot{H}$, where $\\sigma$ is an adjustable parameter describing the strength of the bounce. Motivated by the Newtonian hydrodynamic simulations of \\citet{Bonnerot21_stream_width_evolution}, we set $\\sigma=1$ in this paper. If a small fraction of the orbital kinetic energy is dissipated by nozzle shocks \\citep[as speculated by][]{leloudas16_15lh_TDE}, then it is possible to have $\\sigma >1$, which would make the stream thicker (and self-intersection would occur earlier).\n\n\n\n\nAn example of the stream thickness evolution is shown in Fig. \\ref{fig:H_evolve}, for the geodesic considered in Fig. \\ref{fig:fullorbit}. An interesting result of our simulations is that the positions of nozzle shocks are not at the pericenter or apocenter. This is mainly due to GR apsidal precession \\citep[][see their Figs. 4 and 5]{luminet85_nozzle_shock}. The center of mass of the stream and its upper edge move as test particles on two different orbital planes. The two planes intersect at a line\\footnote{This will later on be referred to as ``Intersection Line 1'' so as to differentiate it with the other ``Intersection Line 2'' between the two (LT-precessed) orbital planes of the stream centers. Both intersection lines are important in our geometric understanding of the physical reason for stream collision. } which goes through the BH. The stream thickness $H$ is dictated by the projected distance of the stream position $\\vec{r}$ to this intersection line, and hence $H=0$ when the stream crosses this line. Due to apsidal precession, it is not possible for this intersection line to go through both the pericenter and apocenter. We find that each orbital winding has two nozzle shocks, one of which is near the pericenter and the other one is far from the pericenter and the apocenter, and that LT precession further causes variations in the nozzle shock positions in each successive winding.\n\n\n\n\n\nIt is important to note a limitation of this model: the effect of nozzle shocks on the stream thickness is treated as an instantaneous rebound when the stream undergoes maximum compression at $H=0$. In reality, pressure forces may continue to be important long after the nozzle shocks, and future works may incorporate the results of pressure forces from hydrodynamic simulations \\citep[see][]{Bonnerot21_stream_width_evolution} into the evolution of stream thickness.\n\n\n\n\n\n\\subsection{Computing stream crossing}\\label{sec:intersect}\n\n\n\\begin{table}\n \\centering\n \\begin{tabular}{c c}\n \\hline\n $i$ & 7\\\\\n $j$ & 6\\\\\n $r$ & 1200.8\\\\\n $\\theta\\rm\\,[rad]$ & 2.60\\\\\n $\\phi\\rm\\,[rad]$ & 4.57\\\\\n $\\Delta s$ & 8.18\\\\\n $H_1+H_2$ & 9.76\\\\\n $\\Delta s\/(H_1+H_2)$ & 0.84\\\\\n $H_2\/H_1$ & 0.77\\\\\n $H_1$ & 5.51\\\\\n $\\cos \\gamma$ & -0.96\\\\\n $v_{\\rm rel}$ & 0.08\\\\\n \\hline\n \\end{tabular}\n \\caption{Properties of the intersection region, including the spatial coordinates ($r,\\, \\theta,\\, \\phi$), minimum separation between the two colliding segments $\\Delta s$, the stream width $H_1$ and $H_2$, the collision angle $\\gamma$, and the relative speed of collision $v_{\\rm rel}$, for the parameters $\\tilde E=0.9999$, $L=6.5$, $\\cos I=0.5$, $a=0.9$ and $\\sigma=1$. All results are in units where $G=M=c=1$.\n }\n \\label{tab:finalresult}\n\\end{table}\n\nTo find the point of stream collision, we first project the 3D orbit onto the $r$-$\\phi$ plane and find the points where the trajectory intersects itself in that 2D projection. These points are labelled as intersection candidates, which are found using the following procedure:\n\n\\begin{enumerate}\n \\item Split the entire orbit into many half-orbit arrays, going from pericenter to apocenter or from apocenter to pericenter.\n \\item Compare the half-orbit array labelled $i=0, 1, 2, \\ldots$, to each half-orbit $j(0$) near the pericenter and the other half have head-on collisions ($\\cos\\gamma<0$) typically further from the pericenter. This is illustrated in Fig. \\ref{fig:modes}, where we show two representative cases of these two modes.\n\nThere is a bimodal distribution of intersection angles (see Fig. \\ref{fig:corplots}), which are either close to 180$^{\\rm o}$ (``head-on'' mode) or 0$^{\\rm o}$ (``rear-end'' mode), with very few cases near 90$^{\\rm o}$. This is expected as we consider a post-Newtonian picture of two elliptical orbits with different orbital planes (due to LT precession) but the same focal point --- the orbits must have two closest approaches and in the limit of high eccentricity, one of the closest approaches is near the pericenter and the other one is far away.\nWhen the collision occurs near the pericenter, the two colliding ends have the same sense of rotation and this leads to $\\cos\\gamma\\simeq 1$. When the intersection point is far away from the pericenter and the apocenter\\footnote{The maximum angular momentum (as required by the tidal disruption of a $1M_\\odot$ star by a $10^6M_\\odot$ BH) gives a minimum apsidal precession angle of about $12^{\\rm o}$, and this means that the intersection point cannot be very close to the apocenter \\citep{dai15_pn1_precession}.}, the nearly radial orbits directly lead to $\\cos\\gamma\\simeq -1$. Later on in \\S \\ref{sec:vrel}, we provide analytical expressions for the radial coordinate of the collision point and the collision angle in the post-Newtonian limit.\n\n\nWe also note that the work by \\citet{guillochon15_dark_year} did not find any ``rear-end'' collisions near the pericenter, and that is because they consider each of the orbital windings to be a series of unconnected ellipses without explicitly resolving the (physically connected) near-pericenter segments between neighboring ellipses --- their setup prevents them from finding the ``rear-end'' collisions that we have identified. As we demonstrate later, the self-crossing shock in the rear-end mode is sufficiently strong to thicken the stream significantly and redistribute the angular momentum of the stream, so we expect rapid circularization of the bound gas due to subsequent shocks.\n\n\n\n\n\n\n\n\\subsection{Intersection radius}\n\nIn about half of the cases we studied, the intersection occurs near the pericenter (the rear-end mode), so the intersection radius is trivial to obtain. For the other half of TDEs in the head-on mode, we find the intersection radius depends mainly on the total angular momentum but less strongly on other parameters. This is illustrated in Fig. \\ref{fig:rint} (it can also be clearly seen on the rightmost panels of Fig. \\ref{fig:corplots}), where we show that the intersection radius $r_{\\rm int}$ generally increases with the total angular momentum $L$, for all BH spins $a$ (different symbol styles) and inclination angles $I$ (different colors). This is caused by weaker apsidal precession at larger angular momentum (or larger pericenter radius). For the mildly relativistic cases where the angular momentum is much larger than the minimum $L_{\\rm min}$, the BH spin and orbital inclination play minor roles in determining the intersection radius. This is because intersection typically occurs between adjacent orbital windings (as detailed in \\S \\ref{sec:delay_time}) and the out-of-plane precession angle\\footnote{Note that $\\Phi_{\\rm LT}$ is defined as the angle between the angular momentum vectors between adjacent windings, which is different from the nodal shift $\\Delta \\Omega_{\\rm nod} \\approx 4\\pi a\/L^3$.} $\\Phi_{\\rm LT}$ (due to the LT effect) is much smaller than the angle of in-plane (apsidal) precession $\\Phi_{\\rm ap}$. In the post-Newtonian approximation, these two angles are given by \\citep[][including the lowest order term due to spin]{merritt13_precession_angles}\n\\begin{eqnarray}\\label{eq:LT_precession}\n \\Phi_{\\rm LT} \\approx {4\\pi a\\sin I\\over L^3},\n\\end{eqnarray}\n\\begin{eqnarray}\\label{eq:apsidal_precession}\n \\Phi_{\\rm ap} \\approx {6\\pi \\over L^2} - {8\\pi a\\cos I \\over L^3}.\n\\end{eqnarray}\nWe note that these two are the angles between the orbital angular momentum vectors ($\\Phi_{\\rm LT}$) and between the orbital periapses $(\\Phi_{\\rm ap})$ in a fixed reference frame, and it can be seen that the in-plane precession is larger than the out-of-plane precession by a factor of $1.5L$ or more.\n\nA polynomial fit to the $\\log(r_{\\rm int})$ values for the $a=0$ cases is given by the following:\n\\begin{multline}\n \\log(r_{\\rm int})=-0.0081L^4+0.253L^3-2.96L^2+15.6L-28.5.\n \\label{eq:fit}\n\\end{multline}\n\nHowever, for the highly relativistic cases with angular momenta only slightly above $L_{\\rm min}$, the intersection radius increases with spin such that $a=-0.9$ (or $0.9$) gives the smallest (largest) $r_{\\rm int}$. This is mainly because the 1.5-order post-Newtonian contribution to apsidal precession, $-8\\pi L^{-3} a \\cos I$ (see Eq. (\\ref{eq:apsidal_precession})), is negative for prograde orbits, and the reduction in apsidal precession angle increases the intersection radius \\citep[see also][]{stone19_TDE_in_GR}. The dependence of $r_{\\rm int}$ on the inclination angle can largely be explained by this effect as well.\n\n\n\n\n\n\\begin{figure}\n\\centering\n\\begin{tabular}{c}\n\\subfloat[$a=-0.9$]{\\includegraphics[width = 0.4\\textwidth]{a=-0.9_N.pdf}}\\\\\n\\subfloat[$a=0.5$]{\\includegraphics[width = 0.4\\textwidth]{a=0.5_N.pdf}}\\\\\n\\subfloat[$a=0.9$]{\\includegraphics[width = 0.4\\textwidth]{a=0.9_N.pdf}}\\\\\n\\end{tabular}\n\\caption{The distribution of orbital winding numbers $N_2$ vs. $N_1$ for three different BH spins, under the full ``loss cone'' assumption. We do not find any prompt collisions (for which $N_1=0$ and $N_2 = 1$) on our grid points for all BH spins.\nFor this case of $\\tilde E=0.9999$, each half-orbit revolution takes 2 months.\n}\n\\label{fig:N12}\n\\end{figure}\n\n\n\\subsection{Delay time}\n\\label{sec:delay_time}\nThe delay between tidal disruption and stream intersection causes a ``dark phase'' before bright emission is generated by shocks and accretion \\citep{dai13_gr_precession, guillochon15_dark_year}. In this subsection, we aim to estimate, in a statistical way, the typical delay time for TDEs of a $1M_\\odot$ star by a $10^6M_\\odot$ BH.\n\nWe denote the half-orbit numbers of the two colliding ends as $N_1=j$ and $N_2=i$, with $N_2>N_1$ by definition. For each BH spin $a$, we simulate about 100 cases (ignoring the plunging ones) of different \\{$L$, $\\cos I$\\}, for each of which we obtain \\{$N_1$, $N_2$\\}. We assigned a statistical weight $w=L$ to each case, which is based on the assumption of the ``full loss cone'' angular momentum distribution and an isotropic distribution of incoming stars \\citep{stone20_tde_rate_review}. The resulting normalized distributions of \\{$N_1$, $N_2$\\} for three different BH spins are shown in Fig. \\ref{fig:N12}. \n\nIn our simulations, the collision typically occurs between consecutive half-orbits, i.e. $N_2=N_1+1$, which is in agreement with \\citet{guillochon15_dark_year}. This can be easily understood as follows. Since the LT precession angle is usually small ($\\ll 1\\rm\\, rad$), the orbital planes of consecutive windings are spatially the closest, which means a collision eventually occurs when the angular width of the stream gradually grows above the angular separation between two adjacent orbital planes (cf. Fig. \\ref{fig:H_evolve}).\n\nWe note that one can tell which mode of intersection the collision follows by looking at $N_1$ and $N_2$: for odd $N_1$ and even $N_2$, the collision occurs in the head-on mode; and for even $N_1$ and odd $N_2$, the collision occurs in the rear-end mode. Fig. \\ref{fig:dpdn1} shows $dP\/dN_1$ for three different BH spins, where $P$ is the probability of occurrence of a TDE with $j=N_1$. For our choice of initial stream width $H_0=5R_\\odot$, we do not find any case of prompt collision for which $N_1 = 0$ on our grid points. This means that less than a few percent of TDEs (those with $\\cos I > 0.95$) may have prompt collisions for any of the three different BH spins ($-0.9, 0.5, 0.9$).\n\n\\begin{figure}\n \\centering\n \\includegraphics[width=3.3in]{dPdn.pdf}\n \\caption{The probability distribution of the smaller winding number of the earlier colliding end, $dP\/dN_1$, for three BH spins, under the full ``loss cone'' assumption. The vertical dashed lines show the median for each distribution. The largest angular momentum considered in our grid is $L=9.62$, for which the apsidal precession angle is about 12 degrees. This means that the orbit has precessed by $90^{\\rm o}$ to be roughly perpendicular to the initial semimajor axis of the incoming star when $N_1\\sim 13$. We show in \\S \\ref{sec:physical_reason} that, under the post-Newtonian approximation, stream collisions occur before this.}\n \n \\label{fig:dpdn1}\n\\end{figure}\n\nWe also find that the delay between tidal disruption and stream intersection is typically between 2 and 7 orbital periods, with longest median delay in the highest spin $a=0.9$ case. We took the orbital energy to be $0.9999$, which corresponds to a semimajor axis of $5000$ gravitational radii and Keplerian period of about 4 months (a half-orbit takes 2 months). This means that the initially self-gravitating part of the fallback stream with initial thickness $H_0=5R_\\odot$ only joins the accretion flow about one to three years after the tidal disruption. A caveat is that the delay time strongly depends on our choice of initial stream width $H_0$ as well as the initial polar angle $\\theta$, but the delay times shown in Fig. \\ref{fig:dpdn1} are representative for most TDEs. This will be discussed in \\S \\ref{sec:physical_reason}.\n\n\\begin{figure}\n \\centering\n \\includegraphics[width = 0.4\\textwidth]{a=-0.9_n1.pdf}\n \\includegraphics[width = 0.4\\textwidth]{a=0.5_n1.pdf}\n \\includegraphics[width = 0.4\\textwidth]{a=0.9_n1.pdf}\n \\caption{The distribution of the orbital winding number $N_1$ for different BH spins: $a=-0.9$ (top panel), $a=0.5$ (middle panel) and $a=0.9$ (bottom panel).}\n \\label{fig:n1}\n\\end{figure}\n\nThe full distribution of $N_1$ as a function of the angular momentum $L$ and inclination $\\cos I$ is shown in Fig. \\ref{fig:n1}, where we find $N_1$ rapidly increases with $L$.\nThis can be qualitatively understood as follows.\nFor a larger $L$, the apsidal precession angle is much smaller and hence the stream thickness increases slower with winding number --- the stream thickness is maximized when the apsidal precession brings the major axis to be nearly perpendicular to \\textit{Intersection Line 1}, which is coincident with the \\textit{initial} major axis according to our initial conditions (as it is defined as the intersection line between the orbital plane of the stream center and that of the upper edge of the stream in the \\textit{initial} orbit). This explains the strong dependence of $N_1$ on $L$. The dependence on the inclination angle $I$ is mainly affected by the LT precession angle $\\Phi_{\\rm LT}$ --- the vertical separation $\\Delta s$ is linearly proportional to $|\\Phi_{\\rm LT}|\\propto |a| \\sin I$ --- a larger inclination angle tends to delay the stream collision. The weak dependence of the apsidal precession angle $\\Phi_{\\rm ap}$ on $I$ also plays a minor role, causing $N_1(I)$ to be non-monotonic for large positive spins $a\\sim 1$ and large inclination angles $I\\gtrsim 45^{\\rm o}$ (see the bottom panel of Fig. 8). A quantitative model for $N_1(L, \\cos I)$ will be described in \\S \\ref{sec:physical_reason}. \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n \n \n \n \n \n \n\n\n\n\n\n\n\n\n\\subsection{Thickness ratio and transverse offset}\nThe hydrodynamics of stream collision strongly depends on the thickness ratio $H_2\/H_1$ as well as the dimensionless transverse offset $\\Delta s\/(H_1+H_2)$ \\citep[e.g.,][]{jiang16_stream_collision}, where $\\Delta s$ is the closest approach distance between the centers of the two colliding geodesics. The ratio $\\Delta s\/(H_1+H_2)$ is an indicator of the fraction of kinetic energy dissipated in the collision -- a higher ratio means less energy dissipation. Subsequently, the results of the stream collision affect the formation of the accretion disk \\citep{bonnerot21_accretion_flow_review}. \n\nIn Figs. \\ref{fig:H21} and \\ref{fig:dsH}, we show $\\log_{10}(H_2\/H_1)$ and $\\Delta s\/(H_1+H_2)$, respectively, for cases with different angular momenta $L$ and inclinations $\\cos I$. The panels are for different BH spins $a=-0.9$, $a=0.5$ and $a=0.9$. In the case of $a=0$, the orbit is equatorial and hence $\\Delta s=0$.\n\n\\begin{figure}\n \\centering\n \\includegraphics[width = 0.4\\textwidth]{a=0_H21.pdf}\n \\includegraphics[width = 0.4\\textwidth]{a=-0.9_H21.pdf}\n \\includegraphics[width = 0.4\\textwidth]{a=0.5_H21.pdf}\n \\includegraphics[width = 0.4\\textwidth]{a=0.9_H21.pdf}\n \\caption{The distribution of thickness ratios $H_2\/H_1$ for different BH spins: $a=0$ (top panel), $a=-0.9$ (second panel), $a=0.5$ (third panel) and $a=0.9$ (bottom panel).}\n \\label{fig:H21}\n\\end{figure}\n\n\\begin{figure}\n \\centering\n \\includegraphics[width = 0.4\\textwidth]{a=-0.9_dsH.pdf}\n \\includegraphics[width = 0.4\\textwidth]{a=0.5_dsH.pdf}\n \\includegraphics[width = 0.4\\textwidth]{a=0.9_dsH.pdf}\n \\caption{The distribution of (dimensionless) transverse separations $\\Delta s\/(H_1+H_2)$ for different BH spins: $a=-0.9$ (top panel), $a=0.5$ (middle panel) and $a=0.9$ (bottom panel).}\n \\label{fig:dsH}\n\\end{figure}\n\n\nWe find that the thickness ratio is close to unity, except for the special cases of nearly plunging angular momenta ($L\\lesssim 5$). This is because the stream thickness is specified by the projected distance from \\textit{Intersection Line 1}, which is very similar for the colliding ends since they are both located at the point of intersection.\n\n\n\n\n\nOn the other hand, we find that the typical transverse offset $\\Delta s$ at the collision point is a small but significant fraction (tens of percent) of the total thickness $H_1 + H_2$, and grazing collisions with $\\Delta s\/(H_1+H_2)\\approx 1$ are very rare. Thus, we expect a large fraction of the gas to be shock-heated in an offset collision, which substantially increases the stream thickness (or directly produces a puffy gaseous envelope). Subsequent secondary shocks will lead to the circularization of the bound gas to form an accretion disk.\n\nWe note that a limitation of our study is that the vertical width $H$ does not provide a full description of the transverse density profile of the stream. The consequence is that a small fraction of the gas (the atmosphere of each stream) may still undergo shock interaction even when $\\Delta s\\gtrsim H_1 + H_2$. It is possible that a weak, oblique shock in a minor collision propagates to the interior of the stream and leads to width expansion. Thus, the criterion for stream collision may actually be somewhat looser than our prescription ($\\Delta s\/(H_1 + H_2)$ somewhat larger than 1) and the delay time shorter. \n\n\n\n\n\n\n\n\n\n\n\n\\subsection{Relative speed of collision}\\label{sec:vrel}\nThe (Lorentz invariant) relative speed of the two colliding streams is given by\n\\begin{eqnarray}\n v_{\\rm rel} = {\\sqrt{(\\vec{p}_1\\cdot \\vec{p}_2)^2 - 1} \\over -\\vec{p}_1\\cdot \\vec{p}_2},\n\\end{eqnarray}\nwhere $\\vec{p}_1$ and $\\vec{p}_2$ are the four-velocities of the two streams at the collision point, whose components can be evaluated in any frame as long as the dot product is calculated using the corresponding metric (we compute the dot product in Boyer-Lindquist coordinates).\nIn Fig. \\ref{fig:rv}, we show the relative speeds for all cases considered in this work. \n\nWe find that the relative velocity strongly depends on the total angular momentum of the stream and that the dependence on the inclination and BH spin is weak.\nThe stream collisions are highly violent in all cases, with relative velocities being a few percent of the speed of light or higher. Note that $v_{\\rm rel}\/2$ better describes the component of the velocity that is dissipated by shocks, because in the Newtonian limit and when all the gas is shock-heated (for $\\Delta s=0$ and $H_1=H_2$), the time-averaged power of shock dissipation is given by\\footnote{The stream collision is expected to be intermittent, because after the segment between the two colliding ends is consumed by shocks, the next episode of collision will be delayed by roughly the duration of the previous collision episode.\nIn the ideal case where the shocked gas does not interact with the cold stream, the shock power alternates between 0 and twice the $\\lara{\\dot{E}}$ given here. }\n$\\lara{\\dot E} = (1\/2)\\dot{M} (v_{\\rm rel}\/2)^{2}=1.9\\times10^{43}\\mathrm{\\, erg\\,s^{-1}}\\, (\\dot{M}\/3\\,M_\\odot\\mathrm{\\,yr^{-1}}) (v_{\\rm rel}\/0.03)^2$, where $\\dot{M}$ is the mass fallback rate. As pointed out by \\citet{jiang16_stream_collision, lu20_self-intersection}, the radiative efficiency of the shocked gas near the collision point is much less than unity (typically a few percent) such that the observed luminosity $L_{\\rm sh}\\ll \\lara{\\dot E}$. We see that the shock-powered emission typically falls short of explaining many TDEs with peak optical luminosity of $\\sim 10^{44}\\rm\\, erg\\,s^{-1}$ \\citep[e.g.,][]{gezari12_ps-10jh} unless the initial orbit of the star is deeply penetrating with pericenter distance much less than the tidal disruption radius.\n\n\\begin{figure}\n \\centering\n \\includegraphics[width = 0.4\\textwidth]{avgrelv.pdf}\n \\includegraphics[width = 0.4\\textwidth]{a=-0.9_rv.pdf}\n \\includegraphics[width = 0.4\\textwidth]{a=0.5_rv.pdf}\n \\includegraphics[width = 0.4\\textwidth]{a=0.9_rv.pdf}\n \\caption{Relative speed of collision in units of $c$. The top plot shows the relative speed for $a=0$, along with the medians of the relative speeds for $a=-0.9$ (second panel), $a=0.5$ (third panel) and $a=0.9$ (bottom panel) over the inclination angles. The solid lines in the top plot illustrate the approximate analytical prediction for the relative speed, as in Eq. (\\ref{eq:relative_velocity}).\n }\n \\label{fig:rv}\n\\end{figure}\n\nRemarkably, we find the relative velocity to be nearly independent of whether the collision is in the head-on or rear-end modes. We demonstrate the physical reason in the following\npost-Newtonian picture. Let us approximate the orbit right before the stream collision as an ellipse with the BH located at one of the focal points. The stream collision is mainly caused by apsidal precession, and hence the collision points in the head-on and rear-end modes are located at true anomaly $\\pi - \\Phi_{\\rm ap}\/2$ and $\\Phi_{\\rm ap}\/2$, respectively, where the apsidal precession angle $\\Phi_{\\rm ap}$ is given by Eq. (\\ref{eq:apsidal_precession}).\n\n\nThese two collision points (which lie on the same line that goes through the BH) are located at radii\n\\begin{eqnarray}\\label{eq:rint}\n r_{\\pm} = {L^2\\over 1 \\pm e_{\\rm orb} \\cos(\\Phi_{\\rm ap}\/2)},\n \\label{eq:rint}\n\\end{eqnarray}\nwhere the squared orbital eccentricity is $e_{\\rm orb}^2 = 1 - L^2\/a_{\\rm orb}$ and the semimajor axis is $a_{\\rm orb} = [2(1-\\tilde E)]^{-1}$. It can be seen that $r_+$ is close to the pericenter radius (corresponding to the rear-end mode) and $r_-$ is typically much larger (the head-on mode). Note that $r_-$ is in agreement with Eq. (7) of \\citet{dai15_pn1_precession}.\n\nConservation of energy and angular momentum along with Eq. (\\ref{eq:rint}) imply that the radial components $v_{\\mathrm{K},r}$ of the Keplerian velocities at these two collision points are the same, given by\n\\begin{eqnarray}\n v_{\\mathrm{K},r}^2 = -{1\\over a_{\\rm orb}} + {1 - e_{\\rm orb}^2 \\cos^2(\\Phi_{\\rm ap}\/2) \\over a_{\\rm orb}(1-e_{\\rm orb}^2)},\n\\end{eqnarray}\nwhere the dependence on the precession angle is $\\cos^2(\\Phi_{\\rm ap}\/2)$ and the sign of $\\cos\\Phi_{\\rm ap}$ does not matter. The velocity component perpendicular to the radial direction is $L\/r$, so the collision angle is given by $\\tan (\\gamma\/2) = r\\,v_{\\mathrm{K},r}\/L$. This shows that in the rear-end (or head-on) collision mode, we typically have $\\gamma\\ll 1\\rm\\, rad$ (or $\\gamma\\sim \\pi$). \nThe relative velocity is twice the radial component of the velocity at the collision points and is given by\n\\begin{eqnarray}\\label{eq:relative_velocity}\n\\begin{split}\n \\left({v_{\\rm rel}\\over 2}\\right)^2\n \n \\approx 2(\\tilde E - 1) + {1 - [1+2L^2(\\tilde E - 1)] \\cos^2(\\Phi_{\\rm ap}\/2) \\over L^2}.\n \n\\end{split}\n\\end{eqnarray}\nIn the limit where $1-e_{\\rm orb}\\ll 1$ and $\\Phi_{\\rm ap} \\lesssim 1\\rm\\, rad$, we obtain the approximation\n\\begin{eqnarray}\n v_{\\rm rel}\\approx {\\Phi_{\\rm ap} \\over L} \\approx {6\\pi \\over L^3} - {8\\pi a\\cos I \\over L^4}.\n \\label{eq:relv}\n\\end{eqnarray}\nThis shows that the relative velocity between the two colliding ends is a power-law function of the total angular momentum ($v_{\\rm rel}\\propto L^{-3}$) or orbital pericenter distance ($\\propto r_{\\rm p,orb}^{-1.5}$) in the weakly relativistic regime. The relative velocity is slightly smaller for larger, prograde BH spins, as illustrated in Fig. \\ref{fig:rv}.\n\n\n\\section{Physical Reason for Intersection}\\label{sec:physical_reason}\nIn this section, we show that the collision is a geometric effect. Throughout this section, we work in the lowest-order post-Newtonian approximation such that $\\Phi_{\\rm LT}\\ll \\Phi_{\\rm ap}$ (see Eqs. \\ref{eq:LT_precession} and \\ref{eq:apsidal_precession}) and assume that collision occurs between successive half-orbits $N_1$ and $N_2=N_1+1$.\n\n\\begin{figure*}\n \\centering\n \\includegraphics[width=0.65\\textwidth]{schematic_gb.pdf}\n \\caption{A schematic picture for the geometric viewpoint of stream collision in the post-Newtonian approximation, where the orientation of an elliptical orbit undergoes apsidal and Lense-Thirring (LT) precessions. A collision occurs either due to the increase in thickness $H$ as the major axis of the orbit moves away from Intersection Line 1 (thick blue line), or due to the decrease in vertical separation between adjacent orbital windings as the major axis become more aligned with Intersection Line 2 (pink solid line).}\n \\label{fig:schematic}\n\\end{figure*}\n\nA schematic picture is shown in Fig. \\ref{fig:schematic}. The initial ($N=0$) orbital plane at the center of mass (COM) of the fallback stream is denoted as $P_{\\rm COM}$, and the particle at the upper edge of the stream lies in the orbital plane labelled as $P_{\\rm e}$. These two planes intersect in a line, which is denoted as \\textit{Intersection Line 1}. The angle between these two planes ($P_{\\rm COM}$ and $P_{\\rm e}$) is given by\n\\begin{eqnarray}\n \\alpha = H_0\/b_{\\rm orb} = {H_0\\sqrt{2(1-\\tilde{E})}\\over L},\n\\end{eqnarray}\nwhere $H_0$ is the initial stream thickness on the minor axis, $b_{\\rm orb} = La_{\\rm orb}^{1\/2}$ is the semi-minor axis, and $a_{\\rm orb} = (2(1-\\tilde{E}))^{-1}$ is the semi-major axis as determined by the specific orbital energy $\\tilde{E}$. For $H_0=5R_\\odot$, $\\tilde E=0.9999$, and $L\\in (6.5, 9.7)$ (the mildly relativistic cases), we find $\\alpha\\sim 5\\times10^{-3}\\rm\\, rad$. The stream thickness $H$ at a given position is related to the inclination angle $\\alpha$ and the projected distance to Intersection Line 1.\nSuppose the collision occurs at radius $r_{\\rm int}$ (as given by Eq. \\ref{eq:rint}) between the $N_1$-th and $(N_1+1)$-th half-orbits. We project the radial vector $\\vec r_{\\rm int}$ onto Intersection Line 1 to obtain the stream thickness in the vertical direction (perpendicular to the orbital plane) as\n\\begin{eqnarray}\\label{eq:stream_width}\n H(N_1)=r_{\\rm int} \\sin\\lrsb{(N_1 + 1)\\Phi_{\\rm ap}\/2} \\alpha.\n\\end{eqnarray}\n\n\\begin{figure}\n\\centering\n\\includegraphics[width = 0.4\\textwidth]{nam09.pdf}\n\\includegraphics[width = 0.4\\textwidth]{na05.pdf}\n\\includegraphics[width = 0.4\\textwidth]{na09.pdf}\n\\caption{Comparison between the analytical prediction for $N_1$ as in Eq. \\ref{eq:collision_criterion} (indicated by dashed or dotted lines) and the numerical calculation of $N_1$ (indicated by points, squares or triangles) for various values of $\\omega_0$ (in different colors) and BH spin: $a=-0.9$ (top panel), $a=0.5$ (middle panel) and $a=0.9$ (bottom panel). We find good overall agreement between the analytical model and the numerical calculations, except for a few cases with $\\cos(I)\\approx 0$ and $\\omega_0\\approx \\pi$ under $a=0.9$ and $L=7$. These correspond to the rare cases of nearly polar orbits (i.e., the star's initial major axis is nearly aligned with the BH spin vector), for which the stream thickness evolution can only be captured by using the full tidal tensor and hence both our analytical\/numerical results may have large errors.}\n\n\n\\label{fig:n1compare}\n\\end{figure}\n\nThe other quantity that determines the occurrence of stream collision is $\\Delta s$, which is the vertical separation between the COMs of the two colliding streams. This can be calculated as follows. Intersection Line $1$ is perpendicular to the initial orbital angular momentum vector $\\vec L$ of the orbit. The black hole spin vector $\\vec S$ makes an inclination angle $I$ with $\\vec L$. The plane defined by $\\vec L$ of the initial orbit and $\\vec S$ is labelled $M_{\\rm i}$. LT precession of $\\vec L$ about $\\vec S$ causes $\\vec L$ to rotate in the direction of $\\vec S\\times\\vec L$, perpendicular to $M_{\\rm i}$. The plane defined by the rotated $\\vec L'$ and $\\vec S$ is labelled by $M_{\\rm f}$. For adjacent orbital windings, the angle between $M_{\\rm i}$ and $M_{\\rm f}$ is $\\Phi_{\\rm LT}$ (eq. \\ref{eq:LT_precession}). $M_{\\rm i}$ and $M_{\\rm f}$ intersect in \\textit{Intersection Line $2$}, which forms an angle $\\omega_0$ with \\textit{Intersection Line $1$}. The angle $\\omega_0\\in (0, \\pi)$ is entirely set by the initial position and angular momentum of the star when it was fed to the TDE loss cone. It is related to the initial polar angle $\\theta_0$ in our Boyer-Lindquist coordinates and the inclination angle $I$ between $\\vec L$ and $\\vec S$ by\n\\begin{eqnarray}\n \\cos(\\theta_0)=\\cos(\\pi-\\omega_0)\\sin(I).\n\\end{eqnarray}\nOur numerical simulations as described in \\S \\ref{sec:methods} and \\ref{sec:parameter_exploration} only considered $\\theta_0=\\pi\/2$ (so as to reduce the dimensions of parameter space), which corresponds to $\\omega_0=\\pi\/2$. The probability density distribution of $\\mr{d} P\/\\mr{d} \\omega_0$ is flat in $(0, \\pi)$, so it is equally likely that $\\omega_0$ is less than or greater than $\\pi\/2$. According to the post-Newtonian picture described in this section, most of the properties of the stream collision (intersection radius $r_{\\rm int}$, thickness ratio $H_2\/H_1$, offset ratio $\\Delta s\/(H_1+H_2)$, relative velocity $v_{\\rm rel}$) do not strongly depend on the choice of $\\omega_0$. However, as we now describe, the delay time is strongly affected by $\\omega_0$.\n\nIn the limit that the cumulative LT precession angle $N_1\\Phi_{\\rm LT}\/2\\ll 1\\rm\\, rad$, the vertical separation between the COMs of the colliding streams is given by\n\\begin{eqnarray}\n \\Delta s'(N_1) = r_{\\rm int} \\sin\\lrsb{(N_1+1)\\Phi_{\\rm ap}\/2 + \\omega_0} \\Phi_{\\rm LT}.\n\\end{eqnarray}\nAs we will see later, the number of half orbits before the collision is $N_1 \\sim 2\/\\Phi_{\\rm ap}$, so cumulative LT precession angle before the stream collision is of the order $\\Phi_{\\rm LT}\/\\Phi_{\\rm ap}\\sim a\\sin I\/L$. Including the cumulative precession of \\textit{Intersection Line 2} before the collision leads to a slightly more accurate expression\n\\begin{eqnarray} \\label{eq:ds}\n \\Delta s(N_1) = r_{\\rm int} \\sin\\lrsb{(N_1+1)\\Phi_{\\rm ap}\/2 + \\omega_0 - N_1\\Phi_{\\rm LT}\/2} \\Phi_{\\rm LT},\n\\end{eqnarray}\nwhich we use (but the two equations above give very similar delay times). Note that as the number of half-orbits $N_1$ increases, the angle between the radial vector $\\vec r_{\\rm int}$ and Intersection Line 2 (the argument in the sine function above) increases. If $\\omega_0 < \\pi\/2$, $\\Delta s$ first increases with $N_1$ and then decreases as the angle between $\\vec r_{\\rm int}$ and Intersection Line 2 approaches $\\pi$. If $\\omega_0>\\pi\/2$, then $\\Delta s$ monotonically decreases with $N_1$.\n\nUsing Eq. \\ref{eq:stream_width} and Eq. \\ref{eq:ds}, the criterion for intersection, $\\Delta s1$. \n\nFor TDEs with small LT precession angles $\\Phi_{\\rm LT}$ (due to small BH spin or large orbital angular momentum) or for the parts of the fallback stream that are initially not self-gravitating (the initial thickness is much larger than $5R_\\odot$), we may have $\\Phi_{\\rm LT}\/\\alpha \\ll 1$. In this regime, stream collision occurs mainly due to the increase of the right-hand side of Eq. (\\ref{eq:collision_criterion}) (i.e. an increase in the stream thickness), and we can estimate the number of half-orbits before the collision to be $N_1 \\simeq 2\\Phi_{\\rm LT}\\sin\\omega_0\/(\\alpha \\Phi_{\\rm ap}) - 1$. On the other hand, for TDEs with large $\\Phi_{\\rm LT}$ or for the very thin self-gravitating parts of the fallback stream, we may have $\\Phi_{\\rm LT}\/\\alpha \\gtrsim 1$. In this regime, stream collision occurs mainly due to the decrease of the left-hand side of Eq. (\\ref{eq:collision_criterion}) (i.e. a decrease in the vertical separation), and the number of half-orbits before the collision can be estimated by $(N_1+1)\\Phi_{\\rm ap}\/2 + \\omega_0 \\simeq \\pi$ or $N_1\\simeq 2(\\pi-\\omega_0)\/\\Phi_{\\rm ap} - 1$. Putting these two regimes together, we obtain the estimate\n\\begin{eqnarray}\\label{eq:analyticN1}\n N_1 \\simeq \\min\\lrsb{{\\Phi_{\\rm LT}\\sin\\omega_0\\over \\alpha}, \\pi - \\omega_0} {2\\over \\Phi_{\\rm ap}} - 1.\n\\end{eqnarray}\n\nFig. \\ref{fig:n1compare} shows a comparison between the analytical prediction for $N_1$ (Eq. \\ref{eq:collision_criterion}) and the numerical results of $N_1$ (based on the algorithm in \\S \\ref{sec:methods}) for various BH spins and three different choices of $\\omega_0=\\pi\/6, \\pi\/2, 5\\pi\/6$. We fix the initial stream thickness to be $H_0=5R_\\odot$ (the same as in previous sections), which means all the cases are in the $\\Phi_{\\rm LT}\/\\alpha \\gtrsim 1$ regime. The key trend in this regime is that $N_1$ is a strong function of the orbital angular momentum, $N_1\\propto L^2$ (the dependence on the inclination angle is less strong, unless $\\cos I \\simeq 1$, which corresponds to very vanishing LT precession). Another important trend is that $N_1$ decreases with $\\omega_0$, and this is because for larger $\\omega_0$ it takes a smaller number of orbital windings to align the major axis of the orbit with Intersection Line 2.\n\n\nWe conclude that the physical reason for stream collisions lies in the geometry of Kerr spacetime: collision occurs when the apsidal precession brings the major axis of the elliptical orbit to be nearly parallel with the plane containing the angular momentum vector and the BH spin vector (minimizing the vertical separation between consecutive orbits) and\/or to be nearly perpendicular to the initial major axis (maximizing the stream thickness). An earlier work by \\citet{guillochon15_dark_year} suggests that the collision is caused by the increase in stream thickness with the orbital winding number as a result of possible energy dissipation by nozzle shocks. However, in our model, the stream thickness evolves according to the BH's tidal forces and collision still occurs when the stream does not get significantly inflated at the nozzle shocks, as motivated by the recent simulation of \\citet{Bonnerot21_stream_width_evolution}.\n\nAccording to our model, the delay times for TDE stream collisions span a very wide range from a few months up to a decade, which mainly depend on $H_0$ (initial stream thickness), $L$ (orbital angular momentum), and $\\omega_0$ (the angle between the initial major axis and the initial $\\vec L$-$\\vec S$ plane). Short delay times correspond to the cases that satisfy one of the following: $L\\lesssim 5$ (highly relativistic TDEs), small $\\Phi_{\\rm LT}\/\\alpha$ (thick initial streams or small BH spins), or $\\omega_0$ close to $\\pi$. On the other hand, long delay times are expected for cases with large $\\Phi_{\\rm LT}\/\\alpha$ (thin initial streams) and $\\omega_0$ far from $\\pi$.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Discussion}\\label{sec:discussion}\nIn this section, we briefly discuss a number of limitations of our current model which may be addressed in future works.\n\n\n(1) We only considered mono-energetic geodesics, whereas the realistic bound debris has a spread in orbital energy -- the gas falling back at a later time is less bound and has a larger apocenter radius. This causes the two colliding ends of the stream to have different energies and their orbits to have different apocenter radii. Since most of the collisions occur between successive half-orbits ($N_2 = N_1 + 1$), the fallback times of the two colliding ends differ by at most one orbital period. The intersection radius will only be affected by the energy spread if the collision occurs near the apocenter (i.e., for angular momentum $L\\sim L_{\\rm max}\\simeq 9.7$). This is because the apocenter radius of the later colliding end is substantially larger than that of the first one. Accounting for the energy spread requires a more sophisticated scheme that records the positions of different fluid elements with different orbital energies on a grid of global Boyer-Lindquist time. However, for most of the parameter space studied in this work, the intersection radius is sufficiently far from the apocenter and the orbital energies of the two colliding ends are approximately the same.\n\n(2) The stream thickness evolution is approximated by Eq. (\\ref{eq:tfe}), which is exact only for the Schwarzschild spacetime in the direction perpendicular to the orbital plane. The stream thickness within the orbital plane is strongly affected by the angular momentum spread across the stream \\citep{cheng14_relativistic_tdes}. Its evolution is governed by a different tidal equation with non-inertial terms due to rotation, although the differences from our Eq. (\\ref{eq:tfe}) become negligible at sufficiently large radii where the orbits are nearly radial. Additionally, BH spin causes a mixing between the stream thickness perpendicular to the orbital plane and the in-plane thickness, since the tidal tensor is non-diagonal \\citep{marck83_tidal_tensor} \\citep[see also][]{kesden12_spinning_BH}. Fully capturing the evolution of the cross-sectional shape of the stream requires two dimensional simulations using the full tidal tensor as well as including gas pressure effects, which are beyond the scope of this work. In Appendix \\ref{sec:Newtonian_tidal_equation}, we show the stream thickness evolution under an even simpler, Newtonian tidal equation, and the results are qualitatively similar to those presented in previous sections.\n\n(3) The effects of pressure forces are only crudely accounted for near the nozzle shocks (where $H=0$) by prescribing a perfect bounce of the vertical velocity $\\dot{H}$. In reality, pressure forces may have long-lasting effects on the stream thickness evolution. \\citet{Bonnerot21_stream_width_evolution} found that the irreversible entropy increase (which is proportional to the adiabatic constant $P\/\\rho^{5\/3}$ for monoatomic gas) at the nozzle shock causes the stream thickness to be affected by pressure forces long after the shock occurrence. Their simulation showed that the long-term effects of pressure forces may only change the stream thickness by a factor of order unity.\nThus, we expect our results to be qualitatively unchanged even when gas pressure is included.\n\n\n(4) An important caveat is that the delay time distribution (Figs. \\ref{fig:N12} and \\ref{fig:dpdn1}) depends sensitively on the initial thickness of the fallback stream $H_0$ as well as the angle $\\omega_0$ between the major axis and the $\\vec{L}$-$\\vec{S}$ plane of the initial orbit. As for the initial stream thickness, we took $H_0=5R_\\odot$, corresponding to the densest part of the stream that is initially confined by self-gravity. However, the most bound part of the fallback stream is non-self-gravitating and much less dense, and it has a larger thickness by a factor of a few to 10 \\citep[see][]{Bonnerot21_stream_width_evolution}. For a larger $H_0$, we expect the delay time before the intersection to be shorter (see eq. \\ref{eq:analyticN1}). Even if the most bound part of the stream has a shorter delay time before collision, it is so far unclear how the shocked gas from earlier fallback affects the later, denser parts of the stream. We speculate that, if the orbits of the denser parts are largely unaffected by the surrounding low-density\\footnote{The density ratio between the unshocked stream and the surrounding gas is of the order $(r\/H)^2 \\sim 10^4$.} gas from earlier fallback, then the long delay times (of up to a decade) obtained from our study will lead to delayed mass feeding to the BH. This may potentially explain the late-time X-ray and UV emission \\citep{vanVelzen19_late-time_UV, jonker20_late-time_Xray}, without invoking a viscous timescale of a few years or longer (which is difficult to achieve if the gas circularizes near the tidal disruption radius).\n\n\n\n\n\n\n\\section{Summary}\\label{sec:summary}\n\nWe have developed an algorithm to find the position of stream collision that results from the tidal disruption of a $1 M_{\\odot}$ star by a $10^6 M_{\\odot}$ supermassive BH. This algorithm takes as input the BH spin $a$ and the stream's orbital parameters: the specific energy $\\tilde E$, the total specific angular momentum $L$, the inclination angle $I$ between the initial orbital angular momentum and the BH spin. The longitudinal motion of the stream follows a geodesic in the Kerr spacetime, and the evolution of the transverse size of the stream is decoupled from the longitudinal evolution. Using an approximate tidal equation (including the effects of pressure forces at the nozzle shocks by assuming a perfect bounce), we calculate the evolution of the vertical thickness of the stream along the geodesic. This allows us to determine the occurrence of collision by comparing the stream thickness and the closest approach separation between different windings.\n\nBy performing a parameter space exploration for different $a$, $L$ and $I$ using this algorithm, we study various properties of the stream collision: intersection radius, collision angle, stream thicknesses at collision, transverse separation between the two colliding ends, relative speed at collision, and the delay time between the stellar disruption and stream collision. Our main findings are summarized as follows.\n\n\n\\begin{itemize}\n \\item\n \n Misalignment of BH spin with the orbital angular momentum may lead to delay of the stream collision. The physical reason for the collision is a geometric effect (Fig. \\ref{fig:schematic}). Collision occurs when the major axis of the elliptical orbit is brought to be nearly parallel with the initial $\\vec L$-$\\vec S$ plane (so as to minimize vertical separation between consecutive orbits) or to be nearly perpendicular to the initial major axis right after the disruption (so as to maximize the stream thickness). We calculate the delay time of the collision numerically (fully relativistic) and analytically (in the post-Newtonian limit) and find it to span a wide range from months up to a decade. This means that the thinnest parts of the fallback stream may only join the accretion flow many years after the initial tidal disruption.\n \n \\item Stream self-intersection occurs in two modes: the head-on mode and the rear-end mode. About half of the TDEs are in the head-on mode, where the collision occurs far from the pericenter and the collision angle is close to 180$^{\\rm o}$ (or $\\cos\\gamma\\simeq -1$); the other half are in the rear-end mode, where the intersection occurs near the pericenter with a very small collision angle close to $0^{\\rm o}$ (or $\\cos\\gamma\\simeq 1$). Very few cases have collision angles close to $90^{\\rm o}$.\n \\item Intersection typically occurs between consecutive half orbits, i.e. $N_2 = N_1 + 1$, where $N_1 $ and $N_2$ are the half-orbit numbers of the two colliding ends.\n \\item The intersection radius $r_{\\rm int}$ for the head-on mode generally increases with the total angular momentum but depends less strongly on other parameters (whereas $r_{\\rm int}\\approx r_{\\rm p}$ for the rear-end mode). This is because intersection typically occurs between adjacent orbital windings and the LT precession angle is much smaller than the apsidal precession angle. Larger angular momenta are associated with smaller apsidal precession angles that lead to larger intersection radii. The weak dependence of $r_{\\rm int}$ on the BH spin and orbital inclination is such that larger $a\\cos (I)$ leads to larger intersection radii.\n \n \n \\item The ratio of the thicknesses of the two colliding ends $H_2\/H_1$ is typically of order unity, and the transverse separation $\\Delta s$ is a small but significant fraction of $H_1 + H_2$ (grazing collisions with $\\Delta s\\approx H_1 + H_2$ are rare). This means that the stream undergoes an offset collision which causes a large fraction of the gas to be shock heated. We expect the collision in the head-on mode to generate a puffy envelope of gas which undergoes secondary shocks and subsequently forms an accretion disk. In the rear-end mode, the stream is expected to be significantly broadened, which leads to further collisions in subsequent orbits and the eventual formation of an accretion disk.\n \\item The relative velocity between the two colliding ends is at least a few percent of the speed of light, so the collision is always violent. The relative velocity mainly depends on the total angular momentum of the stream; $v_{\\rm rel}\\propto L^{-3}$ in the weakly relativistic regime. The weak dependence on the BH spin and inclination is such that larger $a\\cos(I)$ leads to smaller relative velocity. The time-averaged energy dissipation rate by the self-crossing shock is of the order $10^{43}\\rm\\, erg\\,s^{-1}$ for angular momentum near the boundary of tidal disruption ($L\\sim L_{\\rm max}\\simeq 9.7$ for a $1M_\\odot$ star and a $10^6M_\\odot$ BH).\n \n\\end{itemize}\n\nThis work provides a realistic picture of the aftermath of stellar tidal disruption by a spinning BH. Our results for the intersection radius, collision angle, thickness ratio, and transverse offset can be used as initial conditions for hydrodynamic studies of the stream collision process, the subsequent formation of an accretion disk, and the associated electromagnetic emission \\citep[e.g.,][]{bonnerot21_first_light}. Our approach can be improved in the future by including orbital energy spread at different parts of the fallback stream, and employing a more realistic stream width evolution as obtained by hydrodynamic studies.\n\n\n\n\n\\vspace{1cm}\n\n\\noindent\n{\\bf Data availability}\nThe data produced in this study will be shared on reasonable request to the authors.\n\n\\vspace{0.5cm}\n\n\\noindent\n{\\it Acknowledgments.} \nGB was supported by the Caltech Summer Undergraduate Research Fellowship. WL was supported by the David and Ellen Lee Fellowship at Caltech and the Lyman Spitzer, Jr. Fellowship at Princeton University. The research of CB was funded by the Gordon and Betty Moore Foundation through Grant GBMF5076. This project has received funding from the European Union's Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement No 836751.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\nQuantum black holes have been intensively investigated for recent\nyears. This interest was mainly stimulated by the development of\nhigh energy physics. One of the most promising open issues of high\nenergy physics is the development of theories with extra\nspace-like dimensions. It was proposed that extra space-like\ndimensions can lower the Planck scale to the TeV region\n\\cite{arkani-hamed, arkani-hamed_2, antoniadis, Randall99}. It is\nexpected that particle collisions with center-of-mass energy above\nthe Planck scale and impact parameter smaller than the horizon\nradius can be a source for producing micro black holes and branes\n\\cite{BH_production, Eardley, Yoshino, Yoshino_2, Berti,\nCavaglia_Jhep, Kanti_09}. (Some critical remarks to this scenario\ncan be found in \\cite{Voloshin, Rychkov04}). So the energy\nsufficient for producing of micro black holes can in principle be\nachievable even for present generation of particle accelerators.\nThe lowering of the Planck scale possibly allows to by-pass the\nhierarchy problem. Identification of the ultraviolet cutoff scale\nwith electroweak energy one $\\Lambda_{EW}$ can lead to radiative\nstability without using additional assumptions such as\nsupersymmetry or technicolor. It is considered that the\nunification of gravity with other gauge interactions at\nelectroweak scales can be obtained in theories with extra\ndimensions. Another important feature of theories with extra\ndimensions is violation of Newton's law of gravitation $\\sim1\/r^2$\nand this deviation can be possibly probed in the nearest feature.\n\nEvolution of micro black holes, formed in high energy collisions\ncan be divided into several stages, so called phases: loss of\ngauge charges and multipole momenta (balding phase), loss of\nangular momentum (spin-down phase) and the last one is so called\nSchwarzschild phase when a black hole loses energy via Hawking\nradiation. The final stage of Schwarzschild phase is not well\nunderstood due to the lack of fully-fledged theory of quantum\ngravity. Several attempts to understand this stage were made with\nthe help of Generalized Uncertainty Principle (GUP) \\cite{adler,\ncavaglia04, scardigli_2}. It was shown that GUP prevents complete\nevaporation of black holes leaving some remnants. Noncommutative\ngeometry was also applied to quantum black holes. The key point in\nthis approach is the noncommutativity inspired modification of\nblack hole's metrics \\cite{Nicolini08}.\n\nAnother line of research that has renewed interest to micro black\nholes, especially with GUP, is cosmology. Recent investigations of\nthe CMB power spectrum have brought an idea about preinflationary\nepoch \\cite{Powell, Wang}. The preinflationary scenario proposed\nin \\cite{scardigli} was based on the micro black hole production\nand a set of first principles such as generalized uncertainty\nprinciple and holographic principle. These principles allow one to\nobtain self-consistent description of the suppression of the CMB\nquadrupole without additional assumptions. As it was pointed out\nthat GUP can lead to the matter domination in the preinflatory\nepoch. In spite of some success of matter dominant preinflation\nthis scenario has some drawbacks and needs further investigation.\n\nThis paper is organized as follows. In the second section we\nconsider modified generalized uncertainty principle with minimal\nposition and momentum and obtain an expression for the black\nhole's temperature. In the third section we calculate entropy and\nheat capacity of black hole. In the fourth section emission rate\nrelation is considered and black hole's evaporation time is\ncalculated. The fifth section contains conclusions and some\nmathematical details are considered in Appendix.\n\n\\section{Uncertainty principle with minimal length and momentum and black hole's temperature}\nRecent studies in string theory brought an idea about\ngeneralization of Heisenberg uncertainty principle \\cite{gross,\nwitten}. Similar suggestion was proposed to reconcile principles\nof Quantum Mechanics and General Relativity \\cite{maggiore,\nScardigli_99}. Generalized Uncertainty Principle (GUP) takes form:\n\\begin{equation}\\label{GUP}\n\\D p\\D x\\geqslant\\frac{\\h}{2}(1+\\B\\D p^2)\n\\end{equation}\nAs it is known such a generalization leads to appearance of\nminimal uncertainty in position, so called minimal length. It was\nshown that gravity-induced decoherence gives rise to similar\nmodification of uncertainty principle but with minimal uncertainty\nfor momentum \\cite{Kay, Kay_CQG, Kay_07}. These two assumptions\nabout minimal uncertainties in position and momentum might be\nmerged in order to obtain modified generalized uncertainty with\nboth minimal length and momentum. We can write uncertainty\nrelation:\n\\begin{equation}\\label{MGUP}\n\\D x\\D p\\geqslant\\frac{\\h}{2}\\left(1+\\A\\D x^2+\\B\\D p^2\\right)\n\\end{equation}\nIt is easy to show that $\\D x\\geqslant\\D\nx_{min}=\\h\\sqrt{\\B}\/(\\sqrt{1-\\h^2\\A\\B})$ and $\\D p\\geqslant\\D\np_{min}=\\h\\sqrt{\\A}\/(\\sqrt{1-\\h^2\\A\\B})$. We note that in order to\nget these minimal uncertainties parameters $\\A$, $\\B$ must be\npositive and $\\h^2\\A\\B<1$. We also remark that uncertainty\nprinciple (\\ref{MGUP}) was introduced after consideration of a\ngedanken experiment for the simultaneous measurement of position\nand momentum of a particle in de Sitter spacetime \\cite{Bambi}.\n\n The uncertainty relation (\\ref{MGUP}) can be obtained if we suppose that\noperators of position and momentum obey deformed commutation\nrelation:\n\\begin{equation}\\label{al_osc}\n[x,p]=i\\h (1+\\A x^2+\\B p^2),\n\\end{equation}\nwhere parameters of deformation $\\alpha$ and $\\beta$ are assumed\nto be positive ($\\alpha,\\beta>0$) and $\\h^2\\A\\B<1$. We note that\ncommutation relation (\\ref{al_osc}) and uncertainty principle\n(\\ref{MGUP}) firstly appeared in quantum group investigations\n\\cite{kempf1}.\n\nWe point out that some troubles appear when one tries to\ngeneralize the algebra (\\ref{al_osc}) for a case of higher\ndimension. Probably the main difficulty is to satisfy closedness\ncondition. It was proposed to generalize commutation relation\n(\\ref{al_osc}) in the following way \\cite{Banerjee, mignemi}:\n\\begin{eqnarray}\\label{SDS_alg}\n[x_i,p_j]=i\\h\\left(\\delta_{ij}+\\A x_ix_j+\\B\np_jp_i+\\sqrt{\\A\\B}(p_ix_j+x_jp_i)\\right).\n\\end{eqnarray}\nWhereas other two commutation relations need to be written in a\nproper way to fulfil the Jacobi identity. It was also proposed the\nrepresentation of operators obeying (\\ref{SDS_alg}). We also point\nout that classical commutation relations that use classical\ncounterpart of (\\ref{SDS_alg}) are well defined (see Appendix).\n\nLet us come back to the uncertainty (\\ref{MGUP}) and rewrite it in\nthe form:\n\\begin{equation}\n\\frac{1}{\\h\\B}\\left(\\D x-\\sqrt{\\D\nx^2\\left(1-\\h^2\\A\\B\\right)-\\h^2\\B}\\right)\\leqslant\\D\np\\leqslant\\frac{1}{\\h\\B}\\left(\\D x+\\sqrt{\\D\nx^2\\left(1-\\h^2\\A\\B\\right)-\\h^2\\B}\\right)\n\\end{equation}\nIt is easy to see that right hand side inequality does not recover\nordinary uncertainty principle in the limit $\\B\\rightarrow 0$ or\n(and) $\\A\\rightarrow 0$. So we obtain\n\\begin{equation}\\label{final_MGup}\n\\D p\\geqslant\\frac{1}{\\h\\B}\\left(\\D x-\\sqrt{\\D\nx^2\\left(1-\\h^2\\A\\B\\right)-\\h^2\\B}\\right)\n\\end{equation}\n\n It was proposed that applying uncertainty principle to a\nblack hole one can get Hawking temperature up to a some\n``calibration'' factor \\cite{adler}. As it was pointed out by\nAdler and collaborators GUP prevents a black hole from complete\nevaporation and temperature catastrophe. That idea attracted a lot\nof attention and was generalized for the case of higher dimension\n\\cite{cavaglia04, scardigli_2, Myung}.\n\nSince evaporated particles appear just outside of the horizon\nsurface it means that uncertainty in position for them can not be\nless than linear size of a black hole. If we assume that the\nuncertainty in position for a particle is less than linear size of\nthe black hole it means that particle emerges inside the black\nhole and can not penetrate through the horizon surface. On the\nother hand if we suppose that the uncertainty in position is\ngreater than linear size of the black hole then the emitted\nparticle appears not on the outside of horizon surface but\nsomewhere beyond it and as a consequence the temperature of the\nblack hole will be different than it should be. To obtain\ntemperature of the black hole we equate uncertainty in position\nfor emitted particle to the double of Schwarzschild radius.\n\\begin{equation}\\label{uncert-rad}\n\\D x=2R_S.\n\\end{equation}\n\n Having used (\\ref{final_MGup}) and supposing that $\\D\nE=c\\D p$ we obtain the uncertainty in the energy of emitted\nparticle.\n\\begin{equation}\\label{uncert_energy}\n\\D E=c\\D p\\geqslant\\frac{c}{\\h\\B}\\left(2R_S-\\sqrt{\n4R^2_S\\left(1-\\h^2\\A\\B\\right)-\\h^2\\B}\\right)\n\\end{equation}\n\nLet us suppose that we deal with ordinary Hawking effect and\nconsider Hawking radiation just outside the event horizon of a\nSchwarzschild black hole. We identify the uncertainty of energy\n$\\D E$ given by the relation (\\ref{uncert_energy}) with the\nthermal energy of an emitted particle. So we can use well-known\nenergy-temperature relation $\\D E=3T$ (We have used thermal energy\nfor photons and here we set Boltzmann constant equal to unity\n$k_B=1$). Taking into account all these remarks and substituting\nthe mass $M$ of a black hole instead of its gravitational radius\n$R_S$ ($R_S=2GM\/c^2$) we obtain relation for the temperature of\nemitted particles\n\\begin{equation}\\label{BH_temp}\nT=\\frac{c}{\\pi\\h\\B}\\left(\\frac{4GM}{c^2}-\\sqrt{\\frac{16G^2M^2}{c^4}\\left(1-\\h^2\\A\\B\\right)-\\h^2\\B}\\right)\n\\end{equation}\nWe remark that in the latter relation we have used ``calibration''\nfactor $\\pi$ instead of $3$ in order to get correct relation for\nHawking temperature when parameters of deformation tend to zero\n($\\A,\\B\\rightarrow 0$). Probably the correct result for the\nHawking temperature can be reproduced if one uses proper\nequipartition law on the horizon surface or just outside it. It\nwas shown that number of degrees of freedom on the horizon surface\nis defined by its surface area \\cite{Padmanabhan}. So it is worth\nexpecting that proper equipartition law leads to correct\n``calibration'' factor. It should be noted that Hawking\ntemperature for the Schwarzschild-AdS black hole with modified GUP\n(\\ref{MGUP}) was obtained in \\cite{Park08}. But in order to\nreproduce correct relation for the temperature of the black hole\none of the deformation parameters was held fixed \\cite{Park08}.\nUnfixing this parameter and taking Schwarzschild metrics instead\nof Schwarzschild-AdS one we arrive at the relation\n(\\ref{BH_temp}). We point out that uncertainty relation\n(\\ref{MGUP}) can be used with Schwarzschild metrics because it can\nbe derived without any assumption about the de Sitter spacetime.\nWe also remark that temperature of Schwarzschild-AdS black hole\nwith generalized uncertainty principle that leads to appearance of\nminimal momentum was firstly considered in \\cite{BolenGERG}.\n\nIt was pointed out \\cite{BolenGERG} that uncertainty principle\ndoes not describe the origin of such effects as semiclassical wave\nscattering or particle tunnelling but only their consequence on\nthe measurement process. To explain the origin of black hole's\nevaporation it is necessary to know the quantum states of the\nblack hole from which the exact uncertainty principle follows. It\nwas suggested that black holes thermodynamics is a low-energy\neffect of small-scale physics. It is known that any theory of\nquantum gravity contains some kind of uncertainty principle that\nreduces to Heisenberg principle at low energies. So black hole\nthermodynamics should not depend too much on the details of\nunderlying quantum gravity theory \\cite{BolenGERG}. This is in\nagreement with Visser's conclusion that Hawking radiation requires\nordinary quantum mechanics and slowly evolving future horizon, so\nto explain black hole thermodynamics quantum gravity principles\nare not necessary \\cite{visser}.\n\n Uncertainty relation (\\ref{MGUP}) gives rise to the existence\nof a minimal mass of a black hole:\n\\begin{equation}\nM_{min}=\\frac{\\h c^2\\sqrt{\\B}}{4G\\sqrt{1-\\h^2\\A\\B}}=\\frac{\ncM^2_{Pl}\\sqrt{\\B}}{4\\sqrt{1-\\h^2\\A\\B}},\n\\end{equation}\nwhere $M_{Pl}=\\sqrt{\\h c\/G}$ is the Planck's mass. So according to\nthe relation (\\ref{BH_temp}) it leads us to the finite temperature\nat the final point of Hawking radiation, which takes following\nform:\n\\begin{equation}\nT_{final}=\\frac{c}{\\pi\\sqrt{\\B}\\sqrt{1-\\h^2\\A\\B}}\n\\end{equation}\n\nOne can see that uncertainty relation (\\ref{MGUP}) causes\nincreasing of black hole's temperature in comparison with relation\n(\\ref{GUP}).\n\nMinimal uncertainty in position given by relation (\\ref{MGUP})\nsimilarly as in case of (\\ref{GUP}) prevents us from a temperature\ncatastrophe. It was proposed that generalized uncertainty\nprinciple (\\ref{GUP}) prevents black holes from complete\nevaporation in the same way as the standard uncertainty principle\nprevents the hydrogen atom from collapse \\cite{cavaglia04}. To\nmake the influence of a minimal uncertainty in momentum clear let\nus investigate temperature as a function of $M$. Taking the\nderivative $\\partial T\/\\partial M$ and equating it to zero one can\nfind that temperature has an extremum point when the mass of a\nblack hole reaches:\n\\begin{equation}\\label{extremal_mass}\nM_{ext}=\\frac{c^2}{4G\\sqrt{\\A\\left(1-\\h^2\\A\\B\\right)}}.\n\\end{equation}\nIt is easy to see that at this point temperature of the black hole\ntakes minimal value:\n\\begin{equation}\\label{minimal_temp}\nT_{min}=\\frac{c\\h\\sqrt{\\A}}{\\pi\\sqrt{1-\\h^2\\A\\B}}\n\\end{equation}\nWhen the black hole's mass is above $M_{ext}$, temperature\n(\\ref{BH_temp}) increases with the mass rise. When the mass is\nbelow $M_{ext}$ its decreasing leads to increasing of temperature.\nSimilar mass-temperature dependence was obtained in case of\nSchwarzschild-AdS black hole with modified generalized principle\n(\\ref{MGUP}). We note that minimal temperature is the consequence\nof a minimal uncertainty in momentum and it can appear for\ndifferent black hole's metrics. Temperature as the function of\nmass is shown in Fig.\\ref{fig1}.\n\n\\begin{figure}\n \\centerline{\\includegraphics[scale=1,clip]{temp_2.eps}}\n \\caption{Temperature of a black hole as a function of mass.\n Temperature and mass are represented in Planck's units. So $m=M\/M_{Pl}$\n denotes relation of black hole's mass to the Planck's mass. Deformation parameters\n are also written in dimensionless form: $\\A'=\\A L^2_{Pl}$ and $\\B'=\\h^2\\B\/L^2_{Pl}$.\n In order to get that the mass of a black hole at the final point of Hawking radiation to be equal\n to the Planck's mass (or $m=1$) when the parameter $\\A'=0$ we set $\\B'=16$.} \\label{fig1}\n\\end{figure}\nIn the case of higher dimensional space one should use\nSchwarzschild-Tangherlini metrics \\cite{Tangherlini,Myers}. The\nhorizon radius of a black hole takes form:\n\\begin{equation}\\label{GR_radius_higher_dimensions}\nR_S=\\left(\\frac{16\\pi G_dM}{(d-2)\\Omega_{d-2}c^2}\\right)^{1\/(d-3)}\n\\end{equation}\nWhere $d$ is the dimension of space-time, $G_d$ is the\n$d$-dimensional gravitational constant and $\\Omega_{d-2}$ is the\n surface area of $d-2$-dimensional unit hypersphere.\nSo the temperature of a $d$-dimensional black hole takes form:\n\\begin{equation}\\label{temp_stringy_high_dim}\nT=\\frac{(d-3)c}{\\pi\\h\\B}\\left[2\\left(\\frac{16\\pi\nG_dM}{(d-2)\\Omega_{d-2}c^2}\\right)^{1\/(d-3)}-\\sqrt{4\\left(\\frac{16\\pi\nG_dM}{(d-2)\\Omega_{d-2}c^2}\\right)^{2\/(d-3)}(1-\\h^2\\A\\B)-\\h^2\\B}\\right]\n\\end{equation}\nAs it was pointed out in \\cite{scardigli_2} generalized\nuncertainty relation (\\ref{GUP}) for the black hole in the case of\nhigher dimensions should be replaced by another one due to the\nfact that distances can not be probed below the Schwarzschild\nradius (\\ref{GR_radius_higher_dimensions}). So according to\n\\cite{scardigli_2} instead of generalized uncertainty principle\n(\\ref{GUP}) in $d=4+n$-dimensional space-time one should write:\n\\begin{equation}\\label{MBH_GUP}\n\\D p\\D x\\geqslant\\frac{\\h}{2}(1+\\gamma\\D p^{\\frac{n+2}{n+1}}),\n\\end{equation}\nwhere $n$ is the number of additional spacelike dimensions. It was\nnoted \\cite{scardigli_2} that at high energies stringy GUP\n(\\ref{GUP}) can give an uncertainty $\\D x$ larger than the size of\na black hole itself. But scenarios with extra dimensions suggest\nthat all gauge interactions apart from gravity live on $3+1$\ndimensional brane \\cite{arkani-hamed, arkani-hamed_2, antoniadis,\nRandall99}. So modification of GUP (\\ref{MBH_GUP}) is valid only\nfor gravitons, whereas for photons and other SM particles emitted\nby a black hole it is necessary to use uncertainty principle\n(\\ref{GUP}) or (and) (\\ref{MGUP}). Taking into account all the\nabove remarks we can interpret the relation\n(\\ref{temp_stringy_high_dim}) as some upper bound for temperature\nof emitted gravitons.\n\n\n\\section{Entropy and heat capacity of a black hole}\nHaving calculated black hole's temperature we can find black\nhole's entropy using well-known thermodynamical relation:\n\\begin{equation}\ndS=\\frac{c^2}{T}dM\n\\end{equation}\nAfter integration we obtain:\n\\begin{eqnarray}\\label{enthropy_general}\nS=\\frac{\\pi c^3}{4\\h\nG\\A}\\left(\\ln\\Big{|}\\frac{4GM-\\sqrt{16G^2M^2(1-\\h^2\\A\\B)-\\h^2c^4\\B}}{\\h\nc^2\\sqrt{\\B}}\\Big{|}+\\right.\\nonumber\n\\\\\n\\left.\\sqrt{1-\\h^2\\A\\B}\\ln\\Big{|}\\frac{\\sqrt{16G^2M^2(1-\\h^2\\A\\B)-\\h^2\nc^4\\B}+4GM\\sqrt{1-\\h^2\\A\\B}}{\\h c^2\\sqrt{\\B}}\\Big{|}\\right)\n\\end{eqnarray}\n The last formula reproduces well-known relation for the\nBekenstein-Hawking entropy in the limit $\\A,\\B\\rightarrow 0$. We\ncan suppose that parameter $\\A$ is small in comparing with $\\B$ so\nwe can expand the right hand side of equation\n(\\ref{enthropy_general}) into the series over a small parameter\n$\\A$.\n\\begin{eqnarray}\\label{entropy_small_a}\nS=\\frac{\\pi}{4\\h c G}\\left(8G^2M^2+2GM\\sqrt{16G^2M^2-\\h^2\nc^4\\B}-\\frac{\\h^2c^4\\B}{2}\\ln\\left(\\frac{4GM+\\sqrt{16G^2M^2-\\h^2\nc^4\\B}}{\\h c^2\\sqrt{\\B}}\\right)\\right.\\nonumber\n\\\\\n-\\frac{\\A}{c^4}\\left[64G^4M^4+GM\\sqrt{16G^2M^2-\\h^2\nc^4\\B}\\left(\\frac{\\h^2c^4\\B}{2}+16G^2M^2\\right)\\right.\\nonumber\n\\\\\n\\left.\\left.+\\frac{\\h^4c^4\\B^2}{2}\\ln\\left(\\frac{4GM+\\sqrt{16G^2M^2-\\h^2\nc^4\\B}}{\\h c^2\\sqrt{\\B}}\\right)\\right]\\right)\n\\end{eqnarray}\nIn the limit $\\A\\rightarrow 0$ we reproduce the expression for the\n entropy in the presence of a minimal length \\cite {adler}. Expression\n(\\ref{entropy_small_a}) shows that including $\\A$-dependent terms\nlead to decreasing of entropy similarly as we have in the case\nwith $\\B$-terms only.\n\nTo calculate the heat capacity of a black hole we use a well-known\nthermodynamical relation:\n\\begin{equation}\nC=T\\frac{\\partial S}{\\partial T}=\\frac{\\partial E}{\\partial T}.\n\\end{equation}\nIn our case temperature and entropy are represented as functions\nof black hole's mass $M$ (or diameter $R$). So we write the heat\ncapacity as a function of mass $M$.\n\\begin{equation}\\label{heat_capacity_general}\nC=\\frac{\\pi\\h\nc^3\\B}{4G}\\frac{\\sqrt{16G^2M^2(1-\\h^2\\A\\B)-\\h^2c^4\\B}}{\\sqrt{16G^2M^2(1-\\h^2\\A\\B)-\\h^2c^4\\B}-4GM(1-\\h^2\\A\\B)}\n\\end{equation}\n\n\\begin{figure}\n \\centerline{\\includegraphics[scale=1,clip]{heat_capacity_m.eps}}\n \\caption{Heat capacity as a function of mass.}\\label{fig2}\n\\end{figure}\n\nLet us investigate the latter relation in details. It is easy to\nmake yourself sure that when mass is below $M_{ext}$ heat capacity\nis negative and it is equal to zero when the mass reaches\n$M_{min}$. When the mass of black hole is above $M_{ext}$ heat\ncapacity is positive and tends to a finite value when mass goes to\ninfinity. So $M_{ext}$ is the discontinuity point for the heat\ncapacity. Heat capacity as a function of mass is represented in\nFig.\\ref{fig2}. Negative heat capacity shows that thermodynamical\nsystem is unstable and tends to decay. Whereas positive heat\ncapacity makes system stable. So at the extremum point we have\nphase transition . One can conclude that the phase transition is\ncaused by the modified GUP (\\ref{MGUP}) but not particular choice\nof black hole's metrics. As it has already been noted modified\nGUP (\\ref{MGUP}) related to commutation relation (\\ref{SDS_alg})\nbut it is known that the choice of commutation relations that give\nthe same uncertainty principle is non unique. So modified GUP\n(\\ref{MGUP}) might be obtained for more general case of\ncommutation relations. We point out that behaviour of heat\ncapacity near the phase transition point is similar to behavior of\nheat capacity in case of the well-known Hawking-Page phase\ntransition \\cite{Hawking83, Majumdar}. Phase transition also\nappears when the modifications of black hole's metrics caused by\nnonlocal effects or noncommutative geometry are taken into account\n\\cite{Nicolini12}. But in that case stable and unstable phases are\nexchanged their places in comparison with our result. We also note\nthat extremum point $M_{ext}$ is an inflection point for entropy.\nEntropy as a function of mass is shown in Fig.\\ref{fig3}.\n\n\\begin{figure}\n \\centerline{\\includegraphics[scale=1,clip]{ent_m2.eps}}\n \\caption{Entropy as a function of mass}\\label{fig3}\n\\end{figure}\n\n\\section{Emission rate equation and evaporation time of a black hole}\nIn this section evaporation of a micro black hole is studied.\nSimilar problem with GUP was firstly considered in work\n\\cite{adler}. This topic was investigated in details in\n\\cite{cavaglia04, scardigli_2}. To examine quantum black hole's\nevaporation different approaches are used. Probably the simplest\none is to make use of Stefan-Boltzmann law for the black-body\nradiation which is based on an assumption that we have a\nthermodynamical system with fixed temperature so we consider\ncanonical ensemble. We note that the Stefan-Boltzmann law gets new\ncorrection caused by deformation of commutation relations\n\\cite{cavaglia04, scardigli_2, Chang}. As it was pointed out in\n\\cite{casdaio, Casadio_2} canonical description for a micro black\nhole is adequate when the energy of emitted particles (e.g.\nphotons, gravitons) is small in comparison with the energy of a\nblack hole itself. If the energy of emitted particles is\ncomparable with the energy of a black hole itself one should use\nmicrocanonical description. When the energy of black hole is large\nin comparison with the energy of emitted particles both of them\ngive the same result. Here we use canonical description,\nmicrocanonical one will be considered elsewhere. We also point out\nthat statistical mechanics in the case of the deformed commutation\nrelations was considered in \\cite{Fityo}.\n\nLet us examine equilibrium radiation caused by a black hole. We\nsuppose that that black hole is placed inside a sphere of given\nradius $R$ and system is considered under a fixed temperature $T$.\nFor generality we consider $D$-dimensional case ($D=3+n$). So the\nenergy of evaporated particles can be calculated as follows\n(photons, gravitons):\n\\begin{equation}\\label{energy_blackbody}\nE=\\frac{g}{(2\\pi\\h)^D}\\int\\frac{d^Dxd^Dp}{1+\\A x^2+\\B\np^2+2\\sqrt{\\A\\B}({\\bf x},{\\bf p})}\\displaystyle\n\\frac{\\varepsilon(p)\\Gamma (p)}{e^{\\varepsilon(p)\/T}-1}\n\\end{equation}\nwhere $\\varepsilon(p)$ is the dispersion relation for emitted\nparticles, $\\Gamma (p)$ is the greybody factor and $g$ is the\ndegeneracy factor. In the case of photons we put\n$\\varepsilon(p)=cp$ and $g=2$. The same dispersion relation can be\nconsidered for gravitons but the degeneracy factor is different.\nWe also suppose that greybody factor is equal to unity\n$\\Gamma(p)=1$ (blackbody radiation). Let us remind that emission\nof gravitons takes place in $d=4+n$ - dimensional bulk spacetime\nwhereas photons and other gauge particles are emitted on\n3+1-dimensional brane.\n\n We note that suppositions we have made do not\nallow to calculate integral exactly (\\ref{energy_blackbody}) even\nif one of the deformation parameters is absent. To calculate it we\nassume that parameters $\\A$ and $\\B$ are small and develop weight\nfunction under integral into the series over deformation\nparameters. So the latter relation can be rewritten in the form:\n\\begin{equation}\nE=\\frac{g}{(2\\pi\\h)^D}\\int d^Dxd^Dp (1-\\A x^2-\\B\np^2-2\\sqrt{\\A\\B}({\\bf x},{\\bf p}))\\displaystyle\n\\frac{cp}{e^{cp\/T}-1}\n\\end{equation}\nAfter integration we obtain:\n\\begin{eqnarray}\nE=\\frac{2g}{(2\\sqrt{\\pi}c\\h)^D}\\frac{D!\\zeta(D+1)}{\\Gamma(D\/2)}\\left[\\left(V-\\frac{D^{1+2\/D}}{D+2}\\left(\\frac{\\Gamma\n\\left(D\/2\\right)}{2\\pi^{D\/2}}\\right)^{2\/D}\\A\nV^{1+2\/D}\\right)T^{D+1}\\right.\\nonumber\n\\\\\n\\left.-\\frac{(D+2)!\\zeta(D+3)}{D!\\zeta(D+1)}\\B VT^{D+3}\\right]\n\\end{eqnarray}\nwhere $V=\\displaystyle 2\\pi^{D\/2}R^D\/(D\\Gamma(D\/2))$ is the volume\nof $D$-dimensional sphere. It should be noted that the energy of\nradiation nonlinearly depends on the volume. In three dimensional\ncase we find:\n\\begin{equation}\nE=\\frac{g}{(2\\pi\\h)^3}\\left[\\left(V-\\frac{3}{5}\\left(\\frac{3}{4\\pi}\\right)^{2\/3}\\A\nV^{5\/3}\\right)\\frac{4\\pi^5}{15 c^3}T^4-\\frac{32\\pi^7}{63c^3}\\B\nVT^6\\right]\n\\end{equation}\nTo obtain emission rate equation we suppose that particles are\nemitted by sphere with radius $R_S$. So the total energy $dE$\nemitted during the period of time $dt$ can be written in the form:\n\\begin{eqnarray}\n\\frac{dE}{dt}=-\\frac{4gc}{(2\\h\nc)^D}\\frac{D!\\zeta(D+1)}{\\left(\\Gamma(D\/2)\\right)^2}\n\\left[\\left(R^{D-1}_S-\\A\nR^{D+1}_S\\right)T^{D+1}-\\frac{(D+2)!\\zeta(D+3)}{D!\\zeta(D+1)}\\B\nR^{D-1}_S\\frac{T^{D+3}}{c^2}\\right]\n\\end{eqnarray}\nWhere $R_S$ is the Schwarzschild radius. For photons in three\ndimensional case we obtain:\n\\begin{equation}\\label{emiss_rate_E}\n\\frac{dE}{dt}=-\\frac{4\\pi^3}{15c^2\\h^3}\\left((R^2_S-\\A\nR^4_S)T^4-\\frac{40}{21}\\frac{\\pi^2\\B}{c^2}R^2_ST^6\\right)\n\\end{equation}\nIn the limit when parameters of deformation tens to zero the last\nequation gives ordinary Stefan-Boltzmann law for surface of a\nblack hole.\n At the final point of Hawking radiation, when the temperature\nand Schwarzschild radius reach $T_{final}$ and $R_{min}$\nrespectively, the emission rate is finite:\n\\begin{equation}\n\\frac{dE}{dt}\\Big|_{final}=\\frac{c^2}{60\\h\\B(1-\\h^2\\A\\B)^4}\\left(\\frac{76}{21}+5\\h^2\\A\\B\\right).\n\\end{equation}\nSimilarly as in the case of GUP \\cite{cavaglia04} at the final\npoint of Hawking radiation emission rate is finite. It was\nsupposed that when the final stage has been reached, the black\nhole evaporates completely by emitting a hard Planck-size quantum\nwith maximum temperature in a finite period of time proportional\nto the Planck's time \\cite{cavaglia04}. At the same time heat\ncapacity (\\ref{heat_capacity_general}) tends to zero when the mass\nreaches $M_{min}$. It means that black hole cannot exchange heat\nwith surroundings at the final point of Hawking radiation.\n\nNow using relation (\\ref{BH_temp}) and representing Schwarzschild\nradius as a function of a black hole's mass we can write equation\nfor the emission rate of a black hole. Since we have calculated\nthe energy of emitted radiation up to the first order over\ndeformation parameters we estimate emission rate up to first order\ntoo. So equation (\\ref{emiss_rate_E}) can be written in the form:\n\\begin{equation}\n\\frac{dM}{dt}=-\\frac{\\h c^4}{3840\\pi\nG^2M^2}\\left(1+60\\A\\frac{G^2M^2}{c^4}+\\frac{11}{336}\\B\\frac{\\h^2\nc^4}{G^2M^2}\\right)\n\\end{equation}\n\n\nHaving integrated the last equation we obtain simple expression\nfor evaporation time of a black hole:\n\\begin{equation}\\label{time_of_evapor}\n\\frac{1}{3}(M^3_{min}-M^3)-12\\frac{\\A\nG^2}{c^4}(M^5_{min}-M^5)-\\frac{11}{336}\\frac{\\h^2\\B\nc^4}{G^2}(M_{min}-M)=-\\frac{\\h c^4}{3840\\pi G^2}t\n\\end{equation}\nOne can see that modification of generalized uncertainty principle\n(\\ref{MGUP}) leads to further decrease in black hole's evaporation\ntime in comparison with GUP (\\ref{GUP}). Evaporation time as a\nfunction of mass is shown in Fig.\\ref{fig4}\n\n\\begin{figure}\n \\centerline{\\includegraphics[scale=1,clip]{Lifetime_m.eps}}\n \\caption{Evaporation time as a function of mass}\\label{fig4}\n\\end{figure}\n\n\n\\section{Conclusions}\nWe have considered microscopic black hole with modified GUP\n(\\ref{MGUP}) leading to the appearance of a minimal length as well\nas minimal momentum. Uncertainty relation allows one to obtain\nthermodynamical functions of a black hole in a very simple way.\nTherefore it was the main motivation to investigate the black\nhole's thermodynamics under a bit more general assumption such as\ngeneralized uncertainty principle \\cite{adler, cavaglia04}.\nModification of GUP (\\ref{MGUP}) gives rise to some new features\nin black hole's thermodynamics. Similarly as in case of GUP\n(\\ref{GUP}) black hole has finite final temperature which is\ncaused by a minimal uncertainty in position. However, in contrast\nto GUP (\\ref{GUP}) modified GUP (\\ref{MGUP}) leads also to minimal\ntemperature (\\ref{minimal_temp}). Similar result was firstly\nobtained in \\cite{Park08} but there Schwarzschild-AdS black hole's\nmetrics was used and a specific choice for one of the deformation\nparameters was made. This minimal temperature causes important\ninfluence on black hole's thermodynamics. The point of minimal\ntemperature is a point of discontinuity for heat capacity. As we\nhave already pointed out when the mass of a black hole is below\n$M_{ext}$ (\\ref{extremal_mass}) heat capacity is negative and it\ntells us about thermodynamical instability. So in this case black\nhole tends to decay. When the black hole mass is above $M_{ext}$\nheat capacity is positive and black hole is thermodynamically\nstable. We also note that although modified GUP (\\ref{MGUP}) is\nrelated to Snyder-de Sitter commutation relations it can obtained\nfor a more general kind of commutation relations. We can conclude\nthat phase transition which appears due to the presence of minimal\nmomentum and it is possible for different black holes metrics.\n\nWe also investigated thermal radiation of a Schwarzschild black\nhole. We obtained the emission rate equation which is based on\nmodified Stefan-Boltzmann law. Then we used it to calculate the\nevaporation time of a Schwarzschild black hole. In comparison with\nordinary GUP (\\ref{GUP}) modified GUP (\\ref{MGUP}) makes\nevaporation time of a black hole shorter. We also point out that\ncalculations of evaporation time are valid only for unstable phase\nof a black hole when its mass is below $M_{ext}$.\n\n\\section{Appendix}\nIn order to obtain correct relation for the black body radiation\nspectrum we should modify Liouville theorem of the classical\nmechanics caused by a deformation of commutation relations. So we\nhave to find an element of phase-space volume that is invariant\nunder time evolution. To do this let us write classical equations\nof motion supposing that position and momentum coordinates are\nobeyed the classical variant of the algebra (\\ref{SDS_alg}). In\norder to obtain the classical commutation relations the standard\nprocedure is used:\n\\begin{equation}\n\\frac{1}{i\\h}[\\hat{A},\\hat{B}]\\Rightarrow \\{A,B\\}\n\\end{equation}\nUnder this assumption deformed commutation relation\n(\\ref{SDS_alg}) takes following form:\n\\begin{eqnarray}\\label{SDS_brackets}\n\\{x_i,p_j\\}=\\delta_{ij}+\\A x_ix_j+\\B p_ip_j+2\\sqrt{\\A\\B}x_jp_i.\n\\end{eqnarray}\nAs it has already been pointed out in the classical case\ncommutation relation (\\ref{SDS_brackets}) forms a well defined\nalgebra. Other two commutation relations takes form\n\\cite{Banerjee, mignemi}:\n\\begin{eqnarray}\n{\\{x_i,x_j\\}=\\B J_{ij}};\\quad \\{p_i,p_j\\}=\\A J_{ij},\\nonumber\n\\end{eqnarray}\nwhere $J_{ij}$ are the components of angular momentum.\n\nHamilton's equations for the time derivatives of position and\nmomentum read:\n\\begin{eqnarray}\n\\dot{x}_i=\\{x_i,H\\}=\\{x_i,x_j\\}\\frac{\\partial H}{\\partial\nx_j}+\\{x_i,p_j\\}\\frac{\\partial H}{\\partial p_j},\n\\\\\n\\dot{p}_i=\\{p_i,H\\}=\\{p_i,x_j\\}\\frac{\\partial H}{\\partial\nx_j}+\\{p_i,p_j\\}\\frac{\\partial H}{\\partial p_j}.\\nonumber\n\\end{eqnarray}\nConsidering evolution of a system during an infinitesimal period\nof time we obtain:\n\\begin{eqnarray}\nx'_i=x_i+\\delta x_i;\n\\\\\np'_i=p_i+\\delta p_i.\n\\end{eqnarray}\nHere\n\\begin{eqnarray}\n\\delta x_i=\\dot{x}_i\\delta t=\\left(\\{x_i,x_j\\}\\frac{\\partial\nH}{\\partial x_j}+\\{x_i,p_j\\}\\frac{\\partial H}{\\partial\np_j}\\right)\\delta t;\n\\\\\n\\delta p_i=\\dot{p}_i\\delta t=\\left(\\{p_i,x_j\\}\\frac{\\partial\nH}{\\partial x_j}+\\{p_i,p_j\\}\\frac{\\partial H}{\\partial\np_j}\\right)\\delta t.\n\\end{eqnarray}\nAfter infinitesimal evolution an element of phase-space volume\nchanges:\n\\begin{equation}\\label{phase_sp_volume}\nd^Dx'd^Dp'=\\Big|\\frac{\\partial(x'_1,\\ldots,x'_D,p'_1,\\ldots,\np'_D)}{\\partial(x_1,\\ldots,x_D,p_1,\\ldots, p_D)}\\Big|d^Dxd^Dp\n\\end{equation}\nFor generality we consider here $D$-dimensional case. For the\nderivatives we have:\n\\begin{eqnarray}\n\\frac{\\partial x'_i}{\\partial x_j}=\\delta_{ij}+\\frac{\\partial\n\\dot{x}_i}{\\partial x_j}\\delta t;\\quad \\frac{\\partial\nx'_i}{\\partial p_j}=\\frac{\\partial \\dot{x}_i}{\\partial p_j}\\delta\nt;\n\\\\\n\\frac{\\partial p'_i}{\\partial x_j}=\\frac{\\partial\n\\dot{p}_i}{\\partial x_j}\\delta t; \\quad \\frac{\\partial\np'_i}{\\partial p_j}=\\delta_{ij}+\\frac{\\partial \\dot{p}_i}{\\partial\np_j}\\delta t.\n\\end{eqnarray}\nWe calculate the Jacobian in the relation (\\ref{phase_sp_volume})\nup to the first order over the infinitesimal time translation\n$\\delta t$. So under this approximation the Jacobian can be\nwritten in the form:\n\\begin{equation}\nJ=\\Big|\\frac{\\partial(x'_1,\\ldots,x'_D,p'_1,\\ldots,\np'_D)}{\\partial(x_1,\\ldots,x_D,p_1,\\ldots,\np_D)}\\Big|=1+\\left(\\frac{\\partial\\dot{x}_i}{\\partial\nx_i}+\\frac{\\partial\\dot{p}_i}{\\partial p_i}\\right)\\delta t\n\\end{equation}\n So we calculate:\n\\begin{eqnarray}\n\\frac{\\partial\\dot{x}_i}{\\partial\nx_i}+\\frac{\\partial\\dot{p}_i}{\\partial\np_i}=\\frac{\\partial}{\\partial x_i}\\left(\\{x_i,x_j\\}\\frac{\\partial\nH}{\\partial x_j}+\\{x_i,p_j\\}\\frac{\\partial H}{\\partial\np_j}\\right)+\\frac{\\partial}{\\partial\np_i}\\left(\\{p_i,x_j\\}\\frac{\\partial H}{\\partial\nx_j}+\\{p_i,p_j\\}\\frac{\\partial H}{\\partial p_j}\\right)\\nonumber\n\\\\\n=\\frac{\\partial}{\\partial x_i}\\big[\\{x_i,x_j\\}\\big]\\frac{\\partial\nH}{\\partial x_j}+\\{x_i,x_j\\}\\frac{\\partial^2 H}{\\partial\nx_i\\partial x_j}+\\frac{\\partial}{\\partial\nx_i}\\big[\\{x_i,p_j\\}\\big]\\frac{\\partial H}{\\partial\np_j}+\\{x_i,p_j\\}\\frac{\\partial^2 H}{\\partial x_i\\partial\np_j}+\\nonumber\n\\\\\n\\frac{\\partial}{\\partial p_i}\\big[\\{p_i,x_j\\}\\big]\\frac{\\partial\nH}{\\partial x_j}+\\{p_i,x_j\\}\\frac{\\partial^2 H}{\\partial\np_i\\partial x_j}+\\frac{\\partial}{\\partial\np_i}\\big[\\{p_i,p_j\\}\\big]\\frac{\\partial H}{\\partial\np_j}+\\{p_i,p_j\\}\\frac{\\partial^2 H}{\\partial p_i\\partial\np_j}\\nonumber\n\\\\\n=2\\left(\\A x_k\\frac{\\partial H}{\\partial p_k}-\\B p_k\\frac{\\partial\nH}{\\partial x_k}+\\sqrt{\\A\\B}\\left[p_k\\frac{\\partial H}{\\partial\np_k}-x_k\\frac{\\partial H}{\\partial x_k}\\right]\\right)\n\\end{eqnarray}\nFor the element of phase space volume we have:\n\\begin{equation}\\label{phase_sp_volume}\nd^Dx'd^Dp'=d^Dxd^Dp\\left[1+2\\left(\\A x_k\\frac{\\partial H}{\\partial\np_k}-\\B p_k\\frac{\\partial H}{\\partial\nx_k}+\\sqrt{\\A\\B}\\left[p_k\\frac{\\partial H}{\\partial\np_k}-x_k\\frac{\\partial H}{\\partial x_k}\\right]\\right)\\delta\nt\\right]\n\\end{equation}\nLet us consider:\n\\begin{eqnarray}\\label{weight_multipl}\n1+\\A x'^2+\\B p'^2+2\\sqrt{\\A\\B}({\\bf x}',{\\bf p}')=1+\\A(x_i+\\delta\nx_i)^2+\\B(p_i+\\delta p_i)^2\\nonumber\n\\\\\n+2\\sqrt{\\A\\B}(x_i+\\delta x_i,p_i+\\delta p_i)\\simeq 1+\\A x^2+\\B\np^2+2\\sqrt{\\A\\B}({\\bf x},{\\bf p})\\nonumber\n\\\\\n+2(\\A(x_i,\\dot{x}_i)+\\B(p_i,\\dot{p}_i)+\\sqrt{\\A\\B}[(x_i,\\dot{p}_i)+(p_i,\\dot{x}_i)])\\delta\nt=(1+\\A x^2+\\B p^2\\nonumber\n\\\\\n+2\\sqrt{\\A\\B}({\\bf x},{\\bf p}))\\left[1+2\\left(\\A x_k\\frac{\\partial\nH}{\\partial p_k}-\\B p_k\\frac{\\partial H}{\\partial\nx_k}+\\sqrt{\\A\\B}\\left[p_k\\frac{\\partial H}{\\partial\np_k}-x_k\\frac{\\partial H}{\\partial x_k}\\right]\\right)\\delta\nt\\right]\n\\end{eqnarray}\nMaking use of relations (\\ref{phase_sp_volume}) and\n(\\ref{weight_multipl}) we conclude that the following weighted\nphase space volume is invariant under infinitesimal time\ntranslations:\n\\begin{equation}\\label{weighted_volume}\n\\frac{d^Dxd^Dp}{1+\\A x^2+\\B p^2+2\\sqrt{\\A\\B}({\\bf x},{\\bf p})}\n\\end{equation}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\n\\section{Supplementary Information}\n\n\\section{Device Details}\n\nWe fabricated our device from an ``electronic grade'' $\\langle100\\rangle$-oriented diamond purchased from Element Six. The diamond is specified to contain fewer than $5$~ppb nitrogen impurities. Nitrogen-vacancy (NV) centers were introduced via irradiation with $2$~MeV electrons at a fluence of $\\sim 1.2\\times 10^{14}$~cm$^{-2}$ followed by annealing at $850^{\\circ}$C for $2$~hours. The NV centers studied in this work are located at a depth of $\\sim47$~$\\mu$m. \n\nThe high-overtone bulk acoustic resonator (HBAR) used in these measurements consists of a $3$~$\\mu$m thick $\\langle002\\rangle$-oriented ZnO film sandwiched between a Ti\/Pt ($25$~nm\/$200$~nm) ground plane and an Al ($250$~nm) top contact. The piezo-electric ZnO film transduces stress waves into the diamond. The diamond then acts as an acoustic Fabry-P\\'{e}rot cavity to create stress standing wave resonances. Fig.~\\ref{fig:s11} shows a network analyzer measurement of the HBAR admittance ($Y_{11}$) plotted as a function of frequency. From this frequency comb, we selected the $\\omega_{\\text{mech}}\/2\\pi=586$~MHz resonance mode that has a $Q$ of $2700$ as calculated by the $Q$-circle method~\\cite{qCircle} and an on-resonance impedance of $18$~$\\Omega$. This mechanical resonance suppresses driving field amplitude noise that is faster than $\\omega_c=\\frac{\\omega_{\\text{mech}}}{2Q}=110$~kHz. A microwave antenna fabricated on the diamond face opposite the ZnO transducer provides gigahertz frequency magnetic fields for conventional magnetic spin control. \n\n\\begin{figure}[ht]\n\\begin{center}\n\\begin{tabular}{c}\n\\includegraphics[width=\\linewidth]{figS11.pdf} \\\\\n\\end{tabular} \n\\end{center}\n\\caption[fig:s11] {Network analyzer measurement of the power admitted to the HBAR. The inset highlights the $\\omega_{\\text{mech}}\/2\\pi=586$~MHz mode used in the measurements.}\n\\label{fig:s11}\n\\end{figure}\n\n\\section{Mechanical Rabi Driving}\n\nThe mechanically driven Rabi oscillations depicted in Fig.~1a of the main text were measured using the pulse sequence shown in Fig.~\\ref{fig:rabiSeq}. As described in detail in Ref.~\\cite{MacQuarrie2015}, the relatively high $Q$ of our mechanical resonance makes it difficult to perform a traditional pulsed Rabi measurement. Instead, a pair of magnetic $\\pi$-pulses resonant with the $\\Ket{0}\\leftrightarrow\\Ket{-1}$ transition and separated by a fixed time $\\tau_{\\text{mag}}$ is swept through a fixed-length mechanical pulse. The mechanical pulse drives the $\\Ket{+1}\\leftrightarrow\\Ket{-1}$ spin transition, and the duration of this interaction is set by the area of the mechanical pulse enclosed between the two $\\pi$-pulses. By knowing the shape of the mechanical pulse, we convert this enclosed area to effective square-pulse units or an ``effective mechanical pulse length.'' Because the mechanical resonator is pulsed in this experiment, we are able to achieve a larger driving field than in the continuous dynamical decoupling (CDD) Ramsey measurements where the mechanical resonator operates in cw mode. \n\n\\begin{figure}[ht]\n\\begin{center}\n\\begin{tabular}{c}\n\\includegraphics[width=\\linewidth]{rabiSeq.pdf} \\\\\n\\end{tabular} \n\\end{center}\n\\caption[fig:rabiSeq]{Pulse sequence used to measure mechanically driven Rabi oscillations.}\n\\label{fig:rabiSeq}\n\\end{figure}\n\n\\section{Mechanically Dressed Hamiltonian}\n\\label{sec:Ham}\n\nAs mentioned in the main text, we work within the $m_I=+1$ sublevel of the $^{14}$N hyperfine manifold. We consider both a static magnetic field $b$ that is aligned along the NV center symmetry axis and subject to fluctuations $\\delta b$ and a mechanical driving field $\\Omega$ that is subject to amplitude fluctuations $\\delta\\Omega$. In the $\\{+1,0,-1\\}\\otimes\\{(m_I=)+\\frac{1}{2},-\\frac{1}{2}\\}$ Zeeman basis, a nearby $^{13}$C nuclear spin weakly couples to an NV center electronic spin through the hyperfine perturbation $H_{C}=A_{\\|}S_{z}I_{z}$ where $S_z$ and $I_z$ are the spin-$1$ and spin-$\\frac{1}{2}$ Pauli matrices, respectively, and $A_{\\|}$ is the coupling strength~\\cite{slichter}. An NV center electronic spin then obeys the Hamiltonian\n\\begin{widetext}\n$H_{LF}=\\begin{pmatrix}\n\\gamma b_{\\Sigma}+\\frac{1}{2}A_{\\|} & 0 & 0 & 0 & \\Omega_{\\Sigma}\\cos(\\omega_{\\text{mech}} t) & 0 \\\\\n0 & \\gamma b_{\\Sigma}-\\frac{1}{2}A_{\\|} & 0 & 0 & 0 & \\Omega_{\\Sigma}\\cos(\\omega_{\\text{mech}} t) \\\\\n0 & 0 & -D & 0 & 0 & 0 \\\\\n0 & 0 & 0 & -D & 0 & 0 \\\\\n\\Omega_{\\Sigma}\\cos(\\omega_{\\text{mech}} t) & 0 & 0 & 0 & -\\gamma b_{\\Sigma}-\\frac{1}{2}A_{\\|} & 0 \\\\\n0 & \\Omega_{\\Sigma}\\cos(\\omega_{\\text{mech}} t) & 0 & 0 & 0 & -\\gamma b_{\\Sigma}+\\frac{1}{2}A_{\\|} \\\\\n\\end{pmatrix}$\n\\end{widetext}\nwhere $b_{\\Sigma}=b+\\delta b$, $\\Omega_{\\Sigma}=\\Omega+\\delta\\Omega$, other parameters are as defined in the main text, and we have not included a magnetic driving field. Applying the rotating wave approximation and transforming into the reference frame rotating at $\\frac{1}{2}\\omega_{\\text{mech}}=\\frac{1}{2}(2\\gamma b+\\Delta)$ gives the Hamiltonian in the rotating frame\n\\begin{widetext}\n$H_{RF}=\n\\begin{pmatrix}\n\\gamma b_{\\Sigma}+\\frac{1}{2}(\\Delta+A_{\\|}) & 0 & 0 & 0 & \\frac{1}{2}\\Omega_{\\Sigma} & 0 \\\\\n0 & \\gamma b_{\\Sigma}+\\frac{1}{2}(\\Delta+A_{\\|}) & 0 & 0 & 0 & \\frac{1}{2}\\Omega_{\\Sigma} \\\\\n0 & 0 & -D & 0 & 0 & 0 \\\\\n0 & 0 & 0 & -D & 0 & 0 \\\\\n\\frac{1}{2}\\Omega_{\\Sigma} & 0 & 0 & 0 & -\\gamma b_{\\Sigma}-\\frac{1}{2}(\\Delta+A_{\\|}) & 0 \\\\\n0 & \\frac{1}{2}\\Omega_{\\Sigma} & 0 & 0 & 0 & -\\gamma b_{\\Sigma}-\\frac{1}{2}(\\Delta-A_{\\|}) \\\\\n\\end{pmatrix}$.\n\\end{widetext}\nDiagonalizing $H_{RF}$ gives the mechanically dressed Hamiltonian whose energies are quoted in the main text: \n\\begin{widetext}\n$H_{D}= \\begin{pmatrix}\n-D & 0 & 0 & 0 & 0 & 0 \\\\\n0 & -D & 0 & 0 & 0 & 0 \\\\\n0 & 0 & -\\frac{1}{2}\\sqrt{\\Omega_{\\Sigma}^2+\\xi_{-}^2} & 0 & 0 & 0 \\\\\n0 & 0 & 0 & \\frac{1}{2}\\sqrt{\\Omega_{\\Sigma}^2+\\xi_{-}^2} & 0 & 0 \\\\\n0 & 0 & 0 & 0 & -\\frac{1}{2}\\sqrt{\\Omega_{\\Sigma}^2+\\xi_{+}^2} & 0 \\\\\n0 & 0 & 0 & 0 & 0 & \\frac{1}{2}\\sqrt{\\Omega_{\\Sigma}^2+\\xi_{+}^2} \\\\\n\\end{pmatrix}$\n\\end{widetext}\nwhere $\\xi_{\\pm}=\\Delta+2\\gamma\\delta b\\pm A_{\\|}$. In the limit $\\Omega_{\\Sigma}=0$, $H_D$ reduces to the undressed Zeeman Hamiltonian in the rotating frame. \n\n\\section{Dressed State Spectroscopy}\n\nFig.~\\ref{fig:seqs}a shows several concatenated instances of the pulse sequence used for our dressed state spectroscopy measurements. In a single instance, the NV center is optically initialized into the $\\Ket{0}$ spin state at which point a reference fluorescence measurement is made of the full-scale NV center photoluminescence. A magnetic $\\pi$-pulse of strength $\\Omega_{\\text{mag}}\/2\\pi\\sim80$~kHz is then applied to drive a conditional spin rotation. Finally, fluorescence readout provides a quantitative measure of the spin population remaining in $\\Ket{0}$. We interleave $n$ instances of this pulse sequence executed in the dressed basis with $n$ instances of this pulse sequence executed in the undressed basis. In a typical experiment $n\\sim10$, giving a total duty cycle time of $\\sim280$~$\\mu$s and mechanical pulse length of $\\sim140$~$\\mu$s. We differentiate between the dressed and undressed signal by routing the counts from our avalanche photodiode to separate counters on our DAQ. This sequence is then repeated as a function of the magnetic detuning $\\Delta_{\\text{mag}}$ from the $\\Ket{0}\\leftrightarrow\\Ket{-1}$ state splitting to produce the data in Fig.~1d of the main text. \n\nThe dressed signal from this measurement is fit to the sum of two Lorentzians \n\\begin{equation}\n\\begin{split}\nP_{D}&=c_D-\\frac{a_{D,1}}{\\left(\\frac{2}{\\Gamma_{D}}\\right)^2\\left(\\omega-\\frac{1}{2}\\sqrt{\\Delta^2+\\Omega^2}-\\frac{1}{2}\\Delta-\\omega_{0,-1}\\right)^2+1} \\\\\n& -\\frac{a_{D,2}}{\\left(\\frac{2}{\\Gamma_{D}}\\right)^2\\left(\\omega+\\frac{1}{2}\\sqrt{\\Delta^2+\\Omega^2}-\\frac{1}{2}\\Delta-\\omega_{0,-1}\\right)^2+1}\n\\end{split}\n\\label{eq:specFit}\n\\end{equation}\nwhere $P_{D}$ is the measured photoluminescence, $c_D$ is a constant background, $\\omega_{0,-1}$ is the undressed $\\Ket{0}\\leftrightarrow\\Ket{-1}$ spin state splitting, $\\Delta$ is the mechanical detuning, $\\Omega$ is the mechanical driving field, $a_{D,i}$ accounts for the depth of the spectral peaks, and $\\Gamma_D$ is the full width at half maximum of the dressed spectral peaks. The undressed signal is simultaneously fit to the Lorentzian \n\\begin{equation}\nP_{UD}=c_{UD}-\\frac{a_{UD}}{\\left(\\frac{2}{\\Gamma_{UD}}\\right)^2(\\omega-\\omega_{0,-1})^2+1}.\n\\end{equation}\nWe then subtract $\\omega_{0,-1}$ from the $x$-axis to plot photoluminescence as a function of $\\Delta_{mag}$ as shown in Fig.~1d of the main text.\n\n\\begin{figure}[ht]\n\\begin{center}\n\\begin{tabular}{c}\n\\includegraphics[width=\\linewidth]{sequences.pdf} \\\\\n\\end{tabular} \n\\end{center}\n\\caption[fig:seqs]{(a) Pulse sequence used for dressed state spectroscopy measurements. (b) Pulse sequence used for $\\{0,p\\}$ qubit CDD Ramsey measurements. (c) Pulse sequence used for $\\{m,p\\}$ qubit CDD Ramsey measurements. }\n\\label{fig:seqs}\n\\end{figure}\n\n\\section{Expression for the Mechanical Detuning}\n\nIn our spectroscopy measurements, we use the relation $\\frac{1}{2}(\\omega_{0,m}+\\omega_{0,p})-\\omega_{0,-1}=\\frac{1}{2}\\Delta$ as a means of zeroing the mechanical detuning. To derive this expression, we begin in the $\\{+1,0,-1\\}$ basis with the Hamiltonian for an NV center subject to both a mechanical driving field and a magnetic driving field resonant with the $\\Ket{0}\\leftrightarrow\\Ket{-1}$ transition. In the doubly rotating reference frame, this can be written\n\\begin{equation}\nH_{RF}=\\begin{pmatrix}\n\\frac{1}{2}\\Delta & 0 & \\frac{1}{2}\\Omega \\\\\n0 & -D-\\Delta_{\\text{mag}} & \\frac{1}{2}\\Omega_{\\text{mag}} \\\\\n\\frac{1}{2}\\Omega & \\frac{1}{2}\\Omega_{\\text{mag}} & -\\frac{1}{2}\\Delta\\\\\n\\end{pmatrix}\n\\label{eq:hamDelta}\n\\end{equation}\nwhere $\\Delta_{\\text{mag}}=-\\frac{1}{2}\\Delta$ for resonant magnetic driving, and $\\Omega_{\\text{mag}}$ is far enough detuned from the $\\Ket{+1}\\leftrightarrow\\Ket{0}$ transition that we can ignore the $\\Bra{+1}H_{RF}\\Ket{0}$ matrix element. \n\nIn the undressed case ($\\Omega=0$, $\\Delta=0$), the energy of the $\\Ket{0}\\leftrightarrow\\Ket{-1}$ splitting in this reference frame is $\\omega_{0,-1}=D$ where we define $\\hbar=1$. With a non-zero mechanical driving field, calculating the eigenvalues of Eq.~\\ref{eq:hamDelta} to first order in $\\frac{\\Omega_{\\text{mag}}}{\\Omega}$ gives energies $\\omega_{0,p}=D+\\frac{1}{2}(\\Delta+\\sqrt{\\Delta^2+\\Omega^2})$ and $\\omega_{0,m}=D+\\frac{1}{2}(\\Delta-\\sqrt{\\Delta^2+\\Omega^2})$. From this we arrive at the desired expression $\\frac{1}{2}(\\omega_{0,m}+\\omega_{0,p})-\\omega_{0,-1}=\\frac{1}{2}\\Delta$. The same expression is obtained when the $^{13}$C coupling is included. \n\n\\section{Ramsey Measurements}\n\n\\subsection{Dressed Ramsey Pulse Sequences}\n\nFig.~\\ref{fig:seqs}b,c show the pulse sequences used for our CDD Ramsey measurements of the $\\{0,p\\}$ and $\\{m,p\\}$ qubits, respectively. Similar to the spectroscopy experiments, the pulse sequences consist of $2n$ sub-instances where each sub-instance is a single measurement. Here, however, $n\\sim 2$, which leads to mechanical pulse lengths and duty cycle lengths similar to those in the spectroscopy experiments. \n\nA single instance of the $\\{0,p\\}$ qubit CDD Ramsey sequence starts with optical initialization into $\\Ket{0}$ and a reference fluorescence measurement. We then apply a magnetic $\\pi\/2$-pulse of strength $\\Omega_{\\text{mag}}\/2\\pi=696\\pm 7$~kHz to populate the $\\{0,p\\}$ subspace. After a free evolution time $\\tau$, we apply a second magnetic $\\pi\/2$-pulse of the same strength to return the spin population to $\\Ket{0}$ where the signal is read out optically. To help visualize the decay, we advance the phase of the second $\\pi\/2$-pulse by $\\omega_{\\text{rot}}\\tau$. Undressed Ramsey measurements are interleaved with the dressed measurements to reduce the power load on our device and provide a simultaneous measurement of the undressed dephasing time $T_{2,\\{0,-1\\}}^*$. This sequence is then repeated as a function of $\\tau$.\n\nThe pulse sequence used for $\\{m,p\\}$ CDD Ramsey measurements is very similar to the $\\{0,p\\}$ Ramsey sequence. For the $\\{m,p\\}$ qubit, however, the $\\pi\/2$-pulses that address the $\\{0,p\\}$ subspace are replaced by double quantum magnetic $\\pi$-pulses of strength $\\Omega_{\\text{mag}}\/2\\pi=1513\\pm8$~kHz that address the $\\{m,p\\}$ subspace~\\cite{mamin2014}. Additionally, the phase of the magnetic pulse that ends the free evolution time is not advanced at $\\omega_{\\text{rot}}\\tau$ for the $\\{m,p\\}$ qubit measurement. In the interest of reducing the power load on our device, we interleave the dressed $\\{m,p\\}$ Ramsey measurements with undressed measurements that execute the same sequence of magnetic pulses. Because this pulse sequence amounts to a $2\\pi$ rotation of the undressed $\\{0,-1\\}$ qubit, the data obtained during these measurements quantify the NV center spin contrast. For each measurement, the average of this undressed trace $\\langle P_{0,ud}\\rangle$ fixes the amplitude in the fitting functions described below. \n\nDuring the $\\{m,p\\}$ qubit measurements, we periodically measure $\\Delta$ spectroscopically and feedback on $b$ to maintain a relatively constant $\\Delta$. Interpolating linear drift between these measurements, we post-select to include only those data sets for which $\\sigma_{\\Delta}\/2\\pi<60$~kHz and $|\\langle\\Delta\\rangle|\/2\\pi<35$~kHz. \n\n\\subsection{Undressed Ramsey Fitting Function}\n\nWe fit the undressed Ramsey data to the expression \n\\begin{equation}\n\\begin{split}\n\\text{Re}[\\rho_{0,-1}]&=c-\\frac{a}{4} e^{-\\frac{\\tau^2}{T_2^{*2}}}\\{\\cos[(\\omega_{\\text{rot}}+\\Delta_{\\text{mag}}+\\frac{1}{2}A_{\\|})\\tau] \\\\\n&+\\cos[(\\omega_{\\text{rot}}+\\Delta_{\\text{mag}}-\\frac{1}{2}A_{\\|})\\tau]\\}\n\\end{split}\n\\end{equation}\nwhere $\\tau$ is the free evolution time, $\\rho_{0,-1}$ is the $\\{0,-1\\}$ coherence, $c$ is a constant background, $a$ is an overall amplitude that accounts for deviations from perfect spin contrast, $T_2^*$ is the inhomogeneous dephasing time, $\\omega_{\\text{rot}}$ is the rate at which we advance the phase of the second $\\pi\/2$-pulse, $\\Delta_{\\text{mag}}$ is the magnetic detuning, and $A_{\\|}$ quantifies coupling to a nearby $^{13}$C nuclear spin. Of these values, $c$, $a$, $T_{2}^*$, $\\Delta_{\\text{mag}}$, and $A_{\\|}$ are free parameters in our fit. We have assumed the $^{13}$C spin is unpolarized. We use the values of $a$ and $c$ returned from the fits to scale the $y$-axes of our plots. \n\n\\subsection{Dressed Ramsey Fitting Function: The $\\{0,p\\}$ Qubit}\n\nIn our CDD Ramsey measurements of the $\\{0,p\\}$ qubit, we tune the magnetic driving field into resonance with the $\\Ket{0}\\leftrightarrow\\Ket{p}$ transition. For the fits, we zero the magnetic detuning midway between the $^{13}$C sublevels $\\Ket{p,(m_{I}=)+\\frac{1}{2}}$ and $\\Ket{p,-\\frac{1}{2}}$. Assuming $\\Delta=0$, our $\\{0,p\\}$ CDD Ramsey signal is then described by the expression\n\\begin{equation}\n\\begin{split}\n\\text{Re}[\\rho_{0,p}]&=c+\\frac{1}{4}e^{-\\frac{\\tau^2}{T_2^{*2}}}\\lbrace a_p \\cos\\left[\\left(\\Delta_{\\text{mag}}+\\omega_{\\text{rot}}\\right)\\tau+\\phi\\right] \\\\\n&+a_m \\cos\\left[\\left(\\Delta_{\\text{mag}}+\\omega_{\\text{rot}}+\\sqrt{\\Omega^2+A_{\\|}^2}\\right)\\tau+\\phi\\right]\\rbrace\n\\end{split}\n\\end{equation}\nwhere $a_m$ is the spin contrast for the $\\{0,m\\}$ qubit, $a_p$ is the spin contrast for the $\\{0,p\\}$ qubit, $\\phi$ is a constant phase offset, and the other parameters are as defined above. We fix the values of $A_{\\|}$ and $\\omega_{\\text{rot}}$, and we vary $c$, $a_i$, $\\phi$, $\\Omega$, and $\\Delta_{\\text{mag}}$ as free parameters in our fitting procedure. Once again, we use the values of $a_m$, $a_p$, and $c$ returned from the fit to scale the $y$-axis in Fig.~2a of the main text. \n\nIt is important to note that because the dressed qubit Larmor frequency does not scale linearly with magnetic field fluctuations, the Ramsey signal does not follow a strictly Gaussian decay. Nevertheless, we fit our data with a Gaussian envelope to aid comparison with the undressed dephasing time. This is a reasonable approximation over the range of mechanical driving fields accessed in this work. \n\n\\subsection{Dressed Ramsey Fitting Function: The $\\{m,p\\}$ Qubit}\n\nFor the $\\{m,p\\}$ qubit dressed under the condition $\\Delta=0$, our CDD Ramsey signal can be described by the expression\n\\begin{equation}\n\\text{Re}[\\rho_{m,p}]=c+\\frac{\\langle P_{0,ud}\\rangle}{2}e^{-\\frac{\\tau^2}{T_2^{*2}}} \\cos\\left[ \\tau\\sqrt{A_{\\|}^2+\\Omega^2}+\\phi\\right]\n\\end{equation}\nwhere $\\langle P_{0,ud}\\rangle$ measures the spin contrast and the other parameters are as described above. To maximally constrain our fitting procedure, we measure $\\langle P_{0,ud}\\rangle$ by interleaving undressed iterations of the CDD Ramsey protocol into the measurement. We allow $c$, $T_{2}^*$, $\\Omega$, and $\\phi$ to vary as free parameters in our fitting procedure. The results of these fits for the measurements shown in Fig.~3b,c of the main text are displayed in Fig.~\\ref{fig:tablePlot} and Fig.~\\ref{fig:tablePlotN}, respectively. \n\n\\begin{figure}[ht]\n\\begin{center}\n\\begin{tabular}{c}\n\\includegraphics[width=\\linewidth]{tablePlotNotBold.pdf} \\\\\n\\end{tabular} \n\\end{center}\n\\caption[fig:tablePlot]{Data and fits for CDD Ramsey measurements of the $\\{m,p\\}$ qubit when $\\Omega$ was actively stabilized. }\n\\label{fig:tablePlot}\n\\end{figure}\n\n\\begin{figure}[ht]\n\\begin{center}\n\\begin{tabular}{c}\n\\includegraphics[width=\\linewidth]{plotTableNotBoldN.pdf} \\\\\n\\end{tabular} \n\\end{center}\n\\caption[fig:tablePlotN]{Data and fits for CDD Ramsey measurements of the $\\{m,p\\}$ qubit when $\\Omega$ was given a Gaussian noise profile. }\n\\label{fig:tablePlotN}\n\\end{figure}\n\nOur CDD Ramsey measurement of a maximally protected $\\Ket{\\downarrow}$ $^{13}$C sublevel (Fig.~3d in the main text) was fit to the function \n\\begin{equation}\n\\begin{split}\n\\text{Re}[\\rho_{m,p}]&=\\frac{\\langle P_{0,ud}\\rangle}{4}\\lbrace\\sqrt{\\frac{\\Omega}{\\sqrt{\\Omega^2+(2\\gamma\\sigma_b)^4\\tau^2}}}\\cos\\left[\\Omega\\tau+\\phi\\right] \\\\\n&+ e^{-\\frac{t^2}{T_{2,\\uparrow}^*2}}\\cos\\left[\\tau\\sqrt{\\Omega^2+4A_{\\|}^2}+\\phi\\right]\\rbrace +c\n\\end{split}\n\\label{eq:highCohere}\n\\end{equation}\nwhere $\\langle P_{0,ud}\\rangle$ fixes the spin contrast, and $\\Omega$, $\\phi$, $T_{2,\\uparrow}^*$, and $c$ were varied as free parameters. A derivation of the non-Gaussian envelope in Eq.~\\ref{eq:highCohere} is given below on page~\\pageref{sndOrder}. \n\nFor all of our $\\{m,p\\}$ qubit Ramsey plots, we use $\\langle P_{0,ud} \\rangle$ and the value of $c$ returned from the fit to scale the $y$-axis. \n\n\\section{Thermal Stability}\n\nAs mentioned above, we intersperse spectral measurements within CDD Ramsey measurements of the $\\{m,p\\}$ qubit. This allows us to feedback on $b$ and maintain a relatively constant $\\Delta$, but these measurements also quantify the thermal drift over the course of the measurement. A histogram of $\\Delta$ extracted from fitting these spectra to Eq.~\\ref{eq:specFit} quantifies drift in the magnetic bias field $\\sigma_{\\Delta}=2\\gamma\\sigma_{\\text{bias}}$. A histogram of $\\omega_{0,-1}$, however, provides information about both the magnetic bias field drift and the thermal drift according to \n\\begin{equation}\n\\sigma_{0,-1}=\\sqrt{\\left(\\gamma\\sigma_{\\text{bias}}\\right)^2+\\left(\\frac{dD}{dT}\\sigma_T\\right)^2}\n\\end{equation}\nwhere $\\sigma_T$ is the standard deviation of normally distributed thermal drift and $\\frac{dD}{dT}=-74\\times 2\\pi$~kHz\/$^{\\circ}$C is the temperature dependence of $D$~\\cite{awschalomThermo,budkerThermo}. The average of $\\sigma_T$ for the power-leveled data that satisfy our post-selection criteria is $0.25\\pm 0.03^{\\circ}$C. Thermal drift on a similar scale can be expected for the $\\{0,p\\}$ qubit measurements. As shown below on page~\\pageref{tempCoh}, fluctuations of this scale would limit the $\\{0,p\\}$ qubit coherence time to $T_{2,\\{0,p\\}}^*=\\frac{\\sqrt{2}}{\\sigma_{T} dD\/dT}=12\\pm 1$~$\\mu$s. \n\n\\section{Modeling Decoherence}\n\n\\subsection{First Order Fluctuations in $\\omega_{i,j}$}\n\nGenerically, first order deviations in the Larmor frequency $\\omega_{i,j}$ take the form $\\delta\\omega_{i,j}=\\alpha\\delta x$ where $\\alpha$ is a constant. If the fluctuation $\\delta x$ follows a Gaussian distribution with standard deviation $\\sigma_x$, an expression for the associated dephasing rate can be found by calculating the weighted average of a distribution of detuned, un-damped Ramsey signals:\n\\begin{equation}\n\\begin{split}\n\\text{Re}[\\rho_{i,j}] & =\\frac{1}{\\sqrt{2\\pi}\\sigma_x}\\int e^{-\\frac{\\delta x^2}{2\\sigma_x^2}}\\cos\\left[(\\omega_{i,j}+\\alpha\\delta x)\\tau\\right]\\text{d}\\delta x \\\\\n& = e^{-\\frac{1}{2}(\\alpha\\sigma_x \\tau)^2}\\cos\\left(\\omega_{i,j}\\tau\\right).\n\\end{split}\n\\label{eq:sigReplace}\n\\end{equation}\nComparing Eq.~\\ref{eq:sigReplace} with an ideal Ramsey signal given by $\\text{Re}[\\rho_{i,j}]=e^{-\\frac{\\tau^2}{T_{2,\\{i,j\\}}^{*2}}}\\cos\\left(\\omega_{i,j}\\tau\\right)$, we see that $T_{2,\\{i,j\\}}^*=\\frac{\\sqrt{2}}{\\alpha \\sigma_x}$ and therefore $\\Gamma_x=\\frac{2\\pi}{T_{2,\\{i,j\\}}^*}=\\sqrt{2}\\pi\\alpha \\sigma_x$. \n\nFor magnetic field fluctuations experienced by the $\\{0,-1\\}$ qubit, $\\alpha\\delta x\\rightarrow \\gamma\\delta b$. We then find $\\gamma\\sigma_b\/2\\pi=(\\sqrt{2}\\pi T_{2,\\{0,-1\\}}^*)^{-1}$ as quoted in the main text. For thermal fluctuations experienced by the $\\{0,p\\}$ qubit, $\\alpha\\delta x\\rightarrow \\frac{\\text{d}D}{\\text{d}T}\\delta T$, and we arrive at $T_{2,\\{0,p\\}}^*=\\frac{\\sqrt{2}}{\\sigma_{T} dD\/dT}$. For the $\\{m,p\\}$ qubit, expanding $\\omega_{m,p}$ to first order in $\\delta b$ gives \n\\begin{equation}\n\\delta\\omega_{m,p;b}=\\frac{2|A_{\\|}|\\gamma\\delta b}{\\sqrt{A_{\\|}^2+\\Omega^2}},\n\\end{equation}\nfrom which we find $\\Gamma_b=2\\sqrt{2}\\kappa|A_{\\|}|\/T_{2,\\{0,-1\\}}^*$ where $\\frac{1}{\\kappa}=\\frac{1}{\\sqrt{2}\\pi}\\sqrt{A_{\\|}^2+\\Omega^2}$. Similarly, expanding $\\omega_{m,p}$ to first order in $\\delta \\Omega$ gives\n\\begin{equation}\n\\delta\\omega_{m,p;\\Omega}=\\frac{\\Omega \\delta\\Omega}{\\sqrt{A_{\\|}^2+\\Omega^2}},\n\\end{equation}\nfrom which we find $\\Gamma_\\Omega=\\kappa\\Omega\\sigma_{\\Omega}$. \n\\label{tempCoh}\n\n\\subsection{Second Order Magnetic Field Fluctuations}\n\\label{sndOrder}\n\nThe decay envelope of a Ramsey measurement is given by the expression $f(\\tau,\\Omega,\\sigma_b,A_{\\|})=|\\langle e^{i\\delta\\phi}\\rangle|$ where $\\delta\\phi$ is the random phase accumulated in a given duty cycle of the measurement~\\cite{ithier2005}. For the $\\{m,p\\}$ qubit in the case when $\\Delta=0$, the Larmor frequency is given by $\\omega_{m,p}=\\sqrt{(\\Omega+\\delta\\Omega)^2+(A_{\\|}+2\\gamma b)^2}$. To second order in $\\delta b$, fluctuations in $\\omega_{m,p}$ from magnetic field fluctuations are then given by \n\\begin{equation}\n\\begin{split}\n\\delta\\omega_{m,p}&=\\frac{\\partial\\omega_{m,p}}{\\partial b}\\Bigr|_{\\delta b=0}\\delta b+\\frac{\\partial^2\\omega_{m,p}}{\\partial b^2}\\Bigr|_{\\delta b=0}\\frac{\\delta b^2}{2}+O(\\delta b^3) \\\\\n&=\\frac{2\\gamma\\delta b(A_{\\|}^3+A_{\\|}\\Omega^2+\\gamma\\delta b\\Omega^2)}{(A_{\\|}^2+\\Omega^2)^{3\/2}}\n\\end{split}.\n\\end{equation}\nThe random phase accumulated is $\\delta\\phi=\\delta\\omega_{m,p}\\tau$. By averaging this phase over a Gaussian distribution of magnetic field fluctuations, we find \n\\begin{equation}\n\\begin{split}\nf(\\tau,\\Omega,\\sigma_b,A_{\\|})&=\\left\\vert\\frac{1}{\\sqrt{2\\pi}\\sigma_b}\\int_{-\\infty}^{\\infty}e^{i\\delta\\omega_{m,p}\\tau}e^{-\\frac{\\delta b^2}{2\\sigma_b^2}}\\text{d}\\delta b\\right\\vert \\\\\n&=\\sqrt{\\beta}e^{-\\frac{2(\\gamma\\sigma_b A_{\\|}\\beta\\tau)^2}{A_{\\|}^2+\\Omega^2}}\n\\end{split}\n\\label{eq:env}\n\\end{equation}\nwhere\n\\begin{equation}\n\\beta(\\tau,\\Omega,\\sigma_b,A_{\\|})\\equiv\\sqrt{\\frac{(A_{\\|}^2+\\Omega^2)^3}{(A_{\\|}^2+\\Omega^2)^3+(2\\gamma\\sigma_b\\Omega)^4\\tau^2}}.\n\\end{equation}\nTo produce the model curves in Fig.~3b,c of the main text, we numerically solve this expression for the value of $\\tau$ such that $f(\\tau,\\Omega,\\sigma_b,A_{\\|})=\\frac{1}{e}$. \n\nWhen $\\Delta=-|A_{\\|}|$, the two $^{13}$C sublevels follow different decay envelopes that can be computed by setting $A_{\\|}\\rightarrow 0$ and $A_{\\|}\\rightarrow 2 A_{\\|}$ in Eq.~\\ref{eq:env}. In the former case, $f(\\tau,\\Omega,\\sigma_b,0)$ reduces to \n\\begin{equation}\n\\begin{split}\nh(\\tau,\\Omega,\\sigma_b)&=\\sqrt{\\frac{\\Omega}{\\sqrt{\\Omega^2+(2\\gamma\\sigma_b)^4\\tau^2}}}\n\\end{split}\n\\end{equation}\nas seen in the main text. For the case of $A_{\\|}\\rightarrow 2 A_{\\|}$, we approximate the decay as Gaussian. The fitting function for Fig.~3d of the main text then becomes\n\\begin{equation}\n\\begin{split}\n\\text{Re}[\\rho_{m,p}]&=c+\\frac{\\langle P_{0,ud}\\rangle}{4}\\lbrace\\sqrt{\\frac{\\Omega}{\\sqrt{\\Omega^2+(2\\gamma\\sigma_b)^4\\tau^2}}}\\cos\\left[\\Omega\\tau+\\phi\\right] \\\\\n&+e^{-\\frac{t^2}{T_{2,\\uparrow}^{*2}}}\\cos\\left[\\tau\\sqrt{\\Omega^2+4 A_{\\|}^2}+\\phi\\right]\\rbrace\n\\end{split}\n\\end{equation}\nwhere only $\\Omega$, $\\phi$, $c$, and $T_{2,\\uparrow}^*$ were allowed to vary as free parameters.\n\nFor simplicity, this derivation of $f(\\tau,\\Omega,\\sigma_b,A_{\\|})$ does not include driving field noise. Including amplitude noise in the mechanical driving field on the scale of our power-leveled measurements produces no noticeable change in the results of the model over the range of mechanical driving fields addressed here. \n\n\\section{Measuring the Voltage Reflected from the HBAR}\n\nWe monitor the mechanical driving field amplitude by tracking the RF power reflected from the mechanical resonator. An RF circulator redirects the reflected power to an RF diode that converts the ac signal to the dc voltage that we measure. As shown in Fig.~\\ref{fig:diodecal}, this measured voltage scales linearly with the mechanical driving field. However, due to the diode's nonzero threshold voltage, that linear dependence has a nonzero intercept. \n\nWe introduce driving field noise to our experiment by periodically shifting the applied power such that the spread of voltages measured by the RF diode over the course of a measurement is normally distributed with a standard deviation of $\\eta \\langle V_R \\rangle$ where $V_R$ is the reflected voltage and $\\eta$ is a constant. Because Fig.~\\ref{fig:diodecal} has a nonzero intercept, such a distribution of voltages will correspond to a Gaussian distribution of driving fields with a standard deviation of $\\sigma_{\\Omega}=(\\langle\\Omega\\rangle+\\alpha)\\eta$ where $\\alpha\/2\\pi=-133\\pm7$~kHz is the ratio of the intercept to the slope for the line of best fit in Fig.~\\ref{fig:diodecal}. \n\n\\begin{figure}[ht]\n\\begin{center}\n\\begin{tabular}{c}\n\\includegraphics[width=\\linewidth]{driveToVolt.pdf} \\\\\n\\end{tabular} \n\\end{center}\n\\caption[fig:diodecal] {Voltage reflected from the mechanical resonator plotted as a function of the mechanical driving field. }\n\\label{fig:diodecal}\n\\end{figure}\n\n\\section{Coherence of the $\\{+1,-1\\}$ Qubit}\n\n\\begin{figure}[ht]\n\\begin{center}\n\\begin{tabular}{c}\n\\includegraphics[width=\\linewidth]{udRamseyFig.pdf} \\\\\n\\end{tabular} \n\\end{center}\n\\caption[fig:t2UD]{(a) Ramsey measurement of the undressed $\\{0,-1\\}$ qubit for the NV center used in the $\\{m,p\\}$ qubit measurements. (b) Fourier spectrum of (a). }\n\\label{fig:t2UD}\n\\end{figure}\n\nWe compare the coherence of the $\\{m,p\\}$ qubit to that of the undressed $\\{+1,-1\\}$ qubit because in each of these qubits both component states are sensitive to magnetic field fluctuations. Directly measuring the dephasing time of the $\\{+1,-1\\}$ qubit at finite field with high precision is a non-trivial task because the measurement becomes sensitive to double quantum pulse infidelities. Instead, we measure $T_2^*$ of the undressed $\\{0,-1\\}$ qubit (Fig.~\\ref{fig:t2UD}) and rely on the fact that for Gaussian magnetic field fluctuations $T_{2,\\{+1,-1\\}}^*=\\frac{1}{2}T_{2,\\{0,-1\\}}^*$. This gives $T_{2,\\{+1,-1\\}}^*=2.7\\pm0.1$~$\\mu$s as quoted in the main text. This same undressed Ramsey measurement also quantifies $|A_{\\|}|\/2\\pi=150\\pm4$~kHz and $\\sigma_b=2.4\\pm0.1$~mG for this NV center.\n\n\\section{Dressed Spectra Through the $\\Ket{+1}\\leftrightarrow\\Ket{0}$ Transition}\n\n\\begin{figure}[hb]\n\\begin{center}\n\\begin{tabular}{c}\n\\includegraphics[width=\\linewidth]{dressedSpectrum.pdf} \\\\\n\\end{tabular} \n\\end{center}\n\\caption[fig:0pSpect]{(a) Dressed state spectrum for a magnetic pulse swept through the undressed $\\Ket{0}\\leftrightarrow\\Ket{-1}$ transition. (b) Dressed state spectrum for a magnetic pulse swept through the undressed $\\Ket{+1}\\leftrightarrow\\Ket{0}$ transition. (c) Pulse sequence used in these measurements.}\n\\label{fig:0pSpect}\n\\end{figure}\n\nFig.~\\ref{fig:0pSpect} shows spectral measurements of the dressed state splitting as measured by sweeping the detuning of a $\\Omega_{\\text{mag}}\/2\\pi=345\\pm4$~kHz magnetic pulse through the resonance of the undressed (a) $\\Ket{0}\\leftrightarrow\\Ket{-1}$ and (b) $\\Ket{+1}\\leftrightarrow\\Ket{0}$ transitions. All three $^{14}$N hyperfine sublevels are visible in the spectra. Because $\\omega_{\\text{mech}}$ is tuned into resonance with the $\\Ket{(m_{s}=)+1,(m_{I}=)+1}\\leftrightarrow\\Ket{-1,+1}$ transition within the $^{14}$N hyperfine manifold, only the $m_{I}=+1$ peak splits into the dressed states $\\Ket{m,+1}$ and $\\Ket{p,+1}$. In these measurements, the HBAR was powered in $3$~$\\mu$s pulses as shown in Fig.~\\ref{fig:0pSpect}c. This reduced the power load on the device and allowed us to reach higher driving fields than we were able to reach in the CDD Ramsey experiments where the mechanical resonator operates in cw mode. \n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\nWireless Sensor Networks (WSNs) have been recognized as a new\ngeneration of ubiquitous computing systems to support a broad range\nof applications, including monitoring, health care and tracking\nenvironmental pollution levels. Minimizing the total energy\nconsumption in both circuit components and RF signal transmission is\na crucial challenge in designing a WSN. Central to this study is to\nfind energy-efficient modulation and coding schemes in the physical\nlayer of a WSN to prolong the sensor lifetime\n\\cite{Cui_GoldsmithITWC0905, HowardEURASIP2006}. For this purpose,\nenergy-efficient modulation\/coding schemes should be simple enough\nto be implemented by state-of-the-art low-power technology, but\nstill robust enough to provide the desired service. Furthermore,\nsince sensor devices frequently switch from sleep mode to active\nmode, modulation and coding circuits should have fast start-up times\n\\cite{Wang_ISLPED2001} along with the capability of transmitting\npackets during a pre-assigned time slot before new sensed packets\narrive. In addition, a WSN needs a powerful channel coding scheme\n(when the distance between nodes exceeds a certain threshold level)\nto protect transmitted data against the unpredictable and harsh\nnature of channels. We refer to these low-complexity and low-energy\nconsumption approaches in WSNs providing proper link reliability as\n\\emph{Green Modulation\/Coding} (GMC) schemes.\n\nThere have been several recent works on the energy efficiency of\nvarious modulation\/ coding schemes in WSNs (see e.g.,\n\\cite{Cui_GoldsmithITWC0905, TangITWC0407, Chouhan_ITWC1009}). Tang\n\\emph{et al.} \\cite{TangITWC0407} compare the power efficiency of\nPPM and FSK in a WSN over fading channels with path-loss without\nconsidering the effect of channel coding. Reference\n\\cite{Sankarasubramaniam2003} investigates the energy efficiency of\nBCH and convolutional codes with FSK for the optimal packet length\nin a point-to-point WSN. It is shown in\n\\cite{Sankarasubramaniam2003} that BCH codes can improve energy\nefficiency compared to the convolutional code for the optimal fixed\npacket size. Liang \\emph{et al.} \\cite{LiangPACRIM2007} investigate\nthe energy efficiency of uncoded FSK modulation in a WSN, where\nmultiple senders transmit their data to a central node in a\nFrequency-Division Multiple Access (FDMA) fashion. Reference\n\\cite{Hanzo_2009} presents the hardware implementation of the\nForward Error Correction (FEC) encoder in IEEE 802.15.4 WSNs, which\nemploys parallel processing to achieve a low processing latency and\nenergy consumption.\n\nMost of the pioneering work on energy-efficient modulation\/coding,\nincluding research in \\cite{TangITWC0407,\nSankarasubramaniam2003}, has focused only on\nminimizing the energy consumption of transmitting one bit, ignoring\nthe effect of bandwidth and transmission time duration. In a\npractical WSN however, it is shown that minimizing the total energy\nconsumption depends strongly on the active mode duration and the\nchannel bandwidth. References \\cite{Cui_GoldsmithITWC0905},\n\\cite{Chouhan_ITWC1009} and \\cite{JamshidICASSP_2010} address this\nissue in a point-to-point WSN, where a sensor node transmits an\nequal amount of data per time unit to a designated sink node. In\n\\cite{Cui_GoldsmithITWC0905}, the authors show that uncoded MQAM is\nmore energy-efficient than uncoded MFSK for short-range\napplications, while using convolutional coded MFSK over AWGN is\ndesirable for longer distances. This line of work is further\nextended in \\cite{Chouhan_ITWC1009} by evaluating the energy\nconsumption per information bit of a WSN for Reed Solomon (RS) Codes\nand various modulation schemes over AWGN channels with path-loss. In\n\\cite{Cui_GoldsmithITWC0905} and \\cite{Chouhan_ITWC1009}, the\nauthors do not consider the effect of multi-path fading.\n\nMore recently, the attention of researchers has been drawn to\ndeploying rateless codes (e.g., Luby Transform (LT) code\n\\cite{LubyFOCS2002}) in WSNs due to the significant advantages of\nthese codes in erasure channels. For instance in\n\\cite{Eckford_ICC2006}, the authors present a scheme for cooperative\nerror control coding using rateless and Low-Density Generator-Matrix\n(LDGM) codes in a multiple relay WSN. However, investigating the\nenergy efficiency of rateless codes in WSNs with low-energy\nmodulations over realistic fading channel models has received little\nattention. To the best of our knowledge, there is no existing\nanalysis on the energy efficiency of rateless coded modulation that\nconsiders the effect of channel bandwidth and active mode duration\non the total energy consumption in a proactive WSN. This paper\naddresses this issue and presents the first in-depth analysis of the\nenergy efficiency of LT codes with FSK, known as green modulation as\ndescribed in \\cite{JamshidICASSP_2010}. The present analysis is\nbased on a realistic model in proactive WSNs operating over a\nRayleigh fading channel with path-loss. In addition, we obtain the\nprobability mass function of the LT code rate and the corresponding\ncoding gain, and study their effects on the energy efficiency of the\nWSN. This study uses the classical BCH and convolutional codes (as\nreference codes), utilized in IEEE standards, for comparative\nevaluation. Numerical results, supported by some experimental setup\non the computation energy, show that the optimized LT coded FSK\nscheme is the most energy-efficient scheme for distance $d$ greater\nthan the threshold level $d_T$. In addition, although the optimized\nuncoded FSK outperforms coded schemes for $d < d_T$, the energy gap\nbetween LT coded and uncoded FSK is negligible for $d < d_T$\ncompared to the other coded schemes. This result comes from the\nsimplicity and flexibility of the LT codes, and suggests that LT\ncodes are beneficial in practical low-power WSNs with dynamic\nposition sensor nodes.\n\nThe rest of the paper is organized as follows. In Section\n\\ref{System_model}, the proactive system model over a realistic\nwireless channel is described. The energy consumption of both\ncircuits and signal transmission of uncoded MFSK modulation is\nanalyzed in Section \\ref{uncoded_MFSK}. Design of LT codes and the\nenergy efficiency of the LT coded MFSK are presented in Section\n\\ref{analysis_Ch4}. Section \\ref{simulation_Ch5} provides some\nnumerical evaluations using some classical channel codes as well as\nrealistic models to confirm our analysis. Also, some design\nguidelines for using LT codes in practical WSN applications are\npresented. Finally in Section \\ref{conclusion_Ch6}, an overview of\nthe results and conclusions are presented.\n\nFor convenience, we provide a list of key mathematical symbols used\nin this paper in Table I. For simplicity of notation, we use the\nsuperscripts `BC', `CC' and `LT' for BCH, convolutional and LT\ncodes, respectively. We use $\\mathcal{E}$, $\\mathcal{P}$ and $T$ for\nenergy, power and time parameters, respectively. Also, we use the\nsubscript ``$c$'' to distinguish coding parameters from the uncoded\nones.\n\n\\begin{table}\n \\label{table234}\n\\caption{List of Notations } \\centering\n \\begin{tabular}{|l|l|}\n \\hline\n $B$:~Channel bandwidth & $b$:~Number of bits in each symbol\\\\\n $d$:~Transmission distance & $\\mathcal{E}_t$:~Energy of transmitted signal\\\\\n $\\mathbb{E}[~.~]$:~Expectation operator & $\\mathcal{E}_{N}$:~Total energy consumption\\\\\n $M$:~Constellation size & $h_{i}$:~Fading channel coefficient\\\\\n $N$:~Number of sensed message & $\\mathcal{L}_d$:~Path loss gain\\\\\n $n$:~Codeword block length & $\\mathcal{O}(x)$:~Output-node degree distribution\\\\\n $P_b$:~Bit error rate & $\\mathcal{P}_c$:~Circuit power consumption\\\\\n $P_R(\\ell)$:~pmf of LT code rate & $\\mathcal{P}_t$:~Power of transmitted signal\\\\\n $R_c$:~Code rate & $T_{ac}$:~Active mode duration\\\\\n $\\eta$:~Path-loss exponent & $T_{tr}$:~Transient mode duration\\\\\n $\\Omega=\\mathbb{E}\\left[\\vert h_{i} \\vert^2\\right]$ & $T_s$:~Symbol duration\\\\\n $\\Upsilon_{c}$:~Coding gain & $\\gamma_{i}$:~Instantaneous SNR \\\\\n \\hline\n \\end{tabular}\n\\end{table}\n\n\n\\section{System Model and Assumptions}\\label{System_model}\nWe consider a proactive wireless sensor system, in which a sensor\nnode transmits an equal amount of data per time unit to a designated\nsink node. Such a proactive sensor system is typical of many\nenvironmental applications such as sensing temperature, humidity and\nlevel of contamination \\cite{Cordeiro_Book2006}. We assume\na non real-time service application where the data transmission between the\nsensor and the sink nodes does not have tight\nconstraint on delay. The sensor and sink\nnodes synchronize with one another and operate in a real-time based\nprocess as depicted in Fig \\ref{fig: Time-Basis}. During\n\\emph{active mode} period $T_{ac}$, the sensed analog signal is\nfirst digitized by an Analog-to-Digital Converter (ADC), and an\n$N$-bit message sequence $\\mathcal{M}_N\\triangleq (m_1,m_2,...,m_N)$\nis generated, where $N$ is assumed to be fixed, and $m_i \\in \\{0,1\n\\}$, $i=1,2,...,N$. The bit stream is then sent to the channel\nencoder. The encoding process begins by dividing the uncoded message\n$\\mathcal{M}_N$ into blocks of equal length denoted by\n$\\mathcal{B}_j \\triangleq (m_{_{(j-1)k+1}},...,m_{jk})$,\n$j=1,...,\\frac{N}{k}$, where $k$ is the length of any particular\n$\\mathcal{B}_j$, and $N$ is assumed to be divisible by $k$. Each\nblock $\\mathcal{B}_j$ is encoded by a pre-determined channel coding\nscheme to generate a coded bit stream $\\mathcal{C}_j \\triangleq\n(a_{_{(j-1)n+1}},...,a_{jn})$, $j=1,...,\\frac{N}{k}$, with block\nlength $n$, where $n$ is either a fixed value (e.g., for block and\nconvolutional codes) or a random variable (e.g., for LT codes).\n\\begin{figure}[t]\n\\centerline{\\psfig{figure=Time_Basis.eps,width=3.55in}} \\caption{A\npractical duty-cycling process in a proactive WSN. } \\label{fig:\nTime-Basis}\n\\end{figure}\nThe coded stream is then modulated by the FSK scheme and transmitted\nto the sink node. Finally, the sensor node returns to sleep mode,\nand all the circuits are shutdown for sleep mode duration $T_{sl}$.\nWe denote $T_{tr}$ as the transient mode duration consisting of the\nswitching time from sleep mode to active mode (i.e., $T_{sl\n\\rightarrow ac}$) plus the switching time from active mode to sleep\nmode (i.e., $T_{ac \\rightarrow sl}$), where $T_{ac \\rightarrow sl}$\nis short enough to be negligible. Under the above considerations,\nthe sensor\/sink nodes have to process one entire $N$-bit message\n$\\mathcal{M}_N$ during $0 \\leq T_{ac} \\leq T_N-T_{tr}$, where $T_N\n\\triangleq T_{tr}+T_{ac}+T_{sl}$ is fixed and $T_{tr} \\approx T_{sl\n\\rightarrow ac}$.\n\nSince sensor nodes in a typical WSN are densely deployed, the\ndistance between nodes is normally short. Thus, the total circuit\npower consumption, defined by $\\mathcal{P}_c \\triangleq\n\\mathcal{P}_{ct}+\\mathcal{P}_{cr}$, is comparable to the RF transmit\npower consumption denoted by $\\mathcal{P}_t$, where\n$\\mathcal{P}_{ct}$ and $\\mathcal{P}_{cr}$ represent the circuit\npower consumptions for the sensor and sink nodes, respectively.\nTaking these into account, the total energy consumption during the\nactive mode period, denoted by $\\mathcal{E}_{ac}$, is given by\n$\\mathcal{E}_{ac}=(\\mathcal{P}_c+\\mathcal{P}_t)T_{ac}$. Also,\nthe power consumption during the\nsleep mode duration $T_{sl}$ is much smaller than the power\nconsumption in the active mode (due to the low sleep mode leakage\ncurrent) to be negligible. As a result, we have the following\ndefinition.\n\n\\begin{defi}\\label{Definition01}\n\\textbf{(Performance Metric):} The energy efficiency, referred to as\nthe performance metric of the proposed WSN, can be measured by the\ntotal energy consumption in each period $T_N$ corresponding to\n$N$-bit message $\\mathcal{M}_N$ as follows:\n\\begin{equation}\\label{total_energy1}\n\\mathcal{E}_N =\n(\\mathcal{P}_c+\\mathcal{P}_t)T_{ac}+\\mathcal{P}_{tr}T_{tr},\n\\end{equation}\nwhere $\\mathcal{P}_{tr}$ is the circuit power consumption during the\ntransient mode period.\n\\end{defi}\nWe use (\\ref{total_energy1}) to investigate and compare the energy\nefficiency of uncoded and coded FSK for various channel coding\nschemes in the subsequent sections.\n\n\n\\textbf{Channel Model:} The choice of low transmission power in WSNs\nresults in several consequences to the channel model. It is shown by\nFriis \\cite{Friis1946} that a low transmission power implies a small\nrange. For short-range transmission scenarios, the root mean square\n(rms) delay spread is in the range of nanoseconds\n\\cite{Karl_Book2005} which is small compared to the symbol duration\n$T_s=16M~\\mu$s obtained from the bandwidth $B=\\frac{M}{T_s}=62.5$ KHz\nin the IEEE 802.15.4 standard, where $M$ is the constellation size of M-ary FSK\n\\cite[pp. 114-115]{Xiong_2006} and \\cite[p. 49]{IEEE_802_15_4_2006}.\nThus, it is reasonable to expect a flat-fading channel model for\nWSNs. In addition, many transmission environments include\nsignificant obstacle and structural interference by obstacles (such\nas wall, doors, furniture, etc), which leads to reduced\nLine-Of-Sight (LOS) components. This behavior suggests a Rayleigh\nfading channel model. Under the above considerations, the channel\nmodel between the sensor and sink nodes is assumed to be Rayleigh\nflat-fading with path-loss. This assumption is used in many works in\nthe literature (e.g., see \\cite{TangITWC0407} for WSNs). For this\nmodel, we assume that the channel is constant during the\ntransmission of a codeword, but may vary from one codeword to\nanother. We denote the fading channel coefficient corresponding to\nsymbol $i$ as $h_{i}$, where the amplitude $\\big\\vert h_{i}\n\\big\\vert$ is Rayleigh distributed with probability density function\n(pdf) $f_{\\vert h_{i}\n\\vert}(r)=\\frac{2r}{\\Omega}e^{-\\frac{r^2}{\\Omega}},~r \\geq 0$, where\n$\\Omega \\triangleq \\mathbb{E}\\left[\\vert h_{i} \\vert^2\\right]$\n\\cite{Proakis2001}.\n\nTo model the path-loss of a link where the transmitter and receiver\nare separated by distance $d$, let denote $\\mathcal{P}_t$ and\n$\\mathcal{P}_r$ as the transmitted and the received signal powers,\nrespectively. For a $\\eta^{th}$-power path-loss channel, the channel\ngain factor is given by $\\mathcal{L}_d \\triangleq\n\\frac{\\mathcal{P}_t}{\\mathcal{P}_r}=M_ld^\\eta \\mathcal{L}_1$, where\n$M_l$ is the gain margin which accounts for the effects of hardware\nprocess variations, background noise and $\\mathcal{L}_1 \\triangleq\n\\frac{(4 \\pi)^2}{\\mathcal{G}_t \\mathcal{G}_r \\lambda^2}$ is the gain\nfactor at $d=1$ meter which is specified by the transmitter and\nreceiver antenna gains $\\mathcal{G}_t$ and $\\mathcal{G}_r$, and\nwavelength $\\lambda$ (e.g., \\cite{Cui_GoldsmithITWC0905}). As a\nresult, when both fading and path-loss are considered, the\ninstantaneous channel coefficient becomes $G_{i} \\triangleq\n\\frac{h_{i}}{\\sqrt{\\mathcal{L}_d}}$. Denoting $x_i(t)$ as the\ntransmitted signal with energy $\\mathcal{E}_{t}$, the received\nsignal at the sink node is given by\n$y_{i}(t)=G_{i}x_{i}(t)+n_{i}(t)$, where $n_{i}(t)$ is AWGN at the\nsink node with two-sided power spectral density given by\n$\\frac{N_{0}}{2}$. Under the above considerations, the instantaneous\nSignal-to-Noise Ratio (SNR) corresponding to symbol $i$ can be\ncomputed as $\\gamma_{i}=\\frac{\\vert G_{i}\\vert^2\n\\mathcal{E}_t}{N_0}$. Under the assumption of a Rayleigh fading\nchannel model, $\\gamma_{i}$ is chi-square distributed with 2 degrees\nof freedom and with pdf\n$f_{\\gamma}(\\gamma_{i})=\\frac{1}{\\bar{\\gamma}}e^{\n-\\frac{\\gamma_{i}}{\\bar{\\gamma}}}$, where $\\bar{\\gamma} \\triangleq\n\\mathbb{E}[\\vert\nG_{i}\\vert^2]\\frac{\\mathcal{E}_t}{N_0}=\\frac{\\Omega}{\\mathcal{L}_d}\\frac{\\mathcal{E}_t}{N_0}$\ndenotes the average received SNR.\n\n\n\n\n\\section{Energy Consumption of Uncoded Scheme}\\label{uncoded_MFSK}\nIn this section, we consider the uncoded M-ary FSK modulation where\n$M$ orthogonal carriers can be mapped into $b \\triangleq \\log_{2}M$\nbits. Among various sinusoidal carrier-based modulation techniques,\nFSK has been found to provide a good compromise between simple radio\narchitecture, low-power consumption, and requirements on linearity\nof the modulation scheme \\cite{JamshidICASSP_2010, Jamshid_IET2010,\nCui_GoldsmithITWC0905}. Also, this scheme is used in some IEEE\nstandards (e.g., \\cite{IEEE_P802_15}). Since we have $b$ bits during\neach symbol period $T_{s}$, we can write\n\\begin{equation}\\label{active1}\nT_{ac}=\\dfrac{N}{b}T_{s}\\stackrel{(a)}{=}\\dfrac{MN}{B\\log_2 M}.\n\\end{equation}\nwhere $(a)$ comes from the bandwidth $B \\approx M\\times\\Delta f $\nwith the minimum carrier separation $\\Delta f=\\frac{1}{T_s}$ for\nMFSK with the non-coherent detector \\cite[pp. 114-115]{Xiong_2006}. It is\nshown in \\cite{JamshidICASSP_2010} that the transmit energy\nconsumption per each symbol for an uncoded MFSK with non-coherent\ndetector is obtained as\n\\begin{eqnarray}\n\\mathcal{E}_t &\\triangleq& \\mathcal{P}_t T_s \\approx \\left[\\left(\n1-(1-P_s)^{\\frac{1}{M-1}}\\right)^{-1}-2 \\right]\\dfrac{\\mathcal{L}_d\nN_0}{\\Omega}\\\\\n\\notag&\\stackrel{(a)}{=}& \\left[\\left(\n1-\\left(1-\\frac{2(M-1)}{M}P_b\\right)^{\\frac{1}{M-1}}\\right)^{-1}-2\n\\right]\\dfrac{\\mathcal{L}_d N_0}{\\Omega}\\\\\n\\label{deriv1}\n\\end{eqnarray}\nwhere $(a)$ comes from the fact that the relationship between the\naverage Symbol Error Rate (SER) $P_s$ and the average Bit Error Rate\n(BER) $P_b$ of MFSK is given by $P_s=\\frac{2(M-1)}{M}P_b$ \\cite[p.\n262]{Proakis2001}. As a result, the output energy consumption of\ntransmitting $N$-bit during $T_{ac}$ of an uncoded MFSK is computed\nfrom (\\ref{active1}) as follows:\n\\begin{eqnarray}\n\\notag \\mathcal{P}_t T_{ac}&=& \\dfrac{T_{ac}}{T_s}\\mathcal{E}_t\n\\approx \\left[\\left(\n1-\\left(1-\\frac{2(M-1)}{M}P_b\\right)^{\\frac{1}{M-1}}\\right)^{-1}-2\n\\right]\\\\\n\\label{energy_trans1}&\\times&\\dfrac{\\mathcal{L}_d N_0}{\\Omega}\n\\dfrac{N}{\\log_2 M}.\n\\end{eqnarray}\nFor the sensor node with uncoded MFSK, we denote the power\nconsumption of frequency synthesizer, filters and power amplifier as\n$\\mathcal{P}_{Sy}$, $\\mathcal{P}_{Filt}$ and $\\mathcal{P}_{Amp}$,\nrespectively. In this case, the circuit power consumption of the\nsensor node with uncoded MFSK can be obtained as\n\\begin{equation}\\label{sensor_power}\n\\mathcal{P}_{ct}=\\mathcal{P}_{Sy}+\\mathcal{P}_{Filt}+\\mathcal{P}_{Amp},\n\\end{equation}\nwhere $\\mathcal{P}_{Amp}=\\alpha \\mathcal{P}_{t}$ with $\\alpha=0.33$\n\\cite{Cui_GoldsmithITWC0905}, \\cite{TangITWC0407}. In addition, the\npower consumption of the sink circuitry with uncoded MFSK scheme can\nbe obtained as\n\\begin{equation}\\label{sink_power}\n\\mathcal{P}_{cr}=\\mathcal{P}_{LNA}+M\\times(\\mathcal{P}_{Filr}+\\mathcal{P}_{ED})+\\mathcal{P}_{IFA}+\\mathcal{P}_{ADC},\n\\end{equation}\nwhere $\\mathcal{P}_{LNA}$, $\\mathcal{P}_{Filr}$, $\\mathcal{P}_{ED}$,\n$\\mathcal{P}_{IFA}$ and $\\mathcal{P}_{ADC}$ denote the power\nconsumption of Low-Noise Amplifier (LNA), filters, envelop detector,\nIF amplifier and ADC, respectively \\cite{JamshidICASSP_2010}. Since,\nthe power consumption during transient mode period $T_{tr}$ is\ngoverned by the frequency synthesizer in both transmitter and the\nreceiver \\cite{Wang_ISLPED2001}, the energy consumption during\n$T_{tr}$ is obtained as $\\mathcal{P}_{tr}T_{tr}=2\n\\mathcal{P}_{Sy}T_{tr}$ \\cite{Cui_GoldsmithITWC0905}. Substituting\n(\\ref{active1}) and (\\ref{energy_trans1}) in (\\ref{total_energy1}),\nthe total energy consumption of an uncoded MFSK for transmitting\n$N$-bit information in each period $T_N$ for a given $P_b$ is\nobtained as\n\\begin{eqnarray}\n\\notag \\mathcal{E}_N &=& (1+\\alpha) \\left[\\left(\n1-\\left(1-\\dfrac{2(M-1)}{M}P_b\\right)^{\\frac{1}{M-1}}\\right)^{-1}-2\n\\right]\\\\\n\\notag&\\times&\\dfrac{\\mathcal{L}_d N_0}{\\Omega} \\dfrac{N}{\\log_2 M}+\n(\\mathcal{P}_{c}-\\mathcal{P}_{Amp})\\dfrac{MN}{B\\log_{2}M}+\\\\\n\\label{energy_totFSK}&&2 \\mathcal{P}_{Sy}T_{tr}.\n\\end{eqnarray}\n\n\nFor energy optimal designs, the impact of channel coding on the\nenergy efficiency of the proposed WSN must be considered as well. It\nis a well known fact that channel coding is a classical approach\nused to improve the link reliability along with the transmitter\nenergy saving due to providing the coding gain \\cite{Proakis2001}.\nHowever, the energy saving comes at the cost of extra energy spent\nin transmitting the redundant bits in codewords as well as the\nadditional energy consumption in the process of encoding\/decoding.\nFor a specific transmission distance $d$, if these extra energy\nconsumptions outweigh the transmit energy saving due to the channel\ncoding, the coded system would not be energy-efficient compared with\nan uncoded system. In the subsequent sections, we will argue the\nabove problem and determine at what distance use of specific channel\ncoding becomes energy-efficient compared to uncoded systems. In\nparticular, we will show in Section \\ref{simulation_Ch5} that the LT\ncoded modulation surpasses this distance constraint in the proposed\nWSN.\n\n\\section{Energy Consumption Analysis of LT Coded Modulation}\\label{analysis_Ch4}\nIn this section, we present the first in-depth analysis on the\nenergy efficiency of LT coded modulation for the proposed proactive\nWSN. To get more insight into how channel coding affects the circuit\nand RF signal energy consumptions in the system, we modify the\nenergy concepts in Section \\ref{uncoded_MFSK}, in particular, the\ntotal energy consumption expression in (\\ref{energy_totFSK}) based\non the coding gain, code rate and the computation energy. We further\npresent the first study on the tradeoff between LT code rate and\ncoding gain required to achieve a certain BER, and the effect of\nthis tradeoff on the total energy consumption of LT coded MFSK for\ndifferent transmission distances.\n\n\\subsection{Energy Efficiency of Coded System}\nFor an arbitrary channel coding scheme, each $k$-bit message\n$\\mathcal{B}_j \\in \\mathcal{M}_N$ is encoded into the codeword\n$\\mathcal{C}_j$ with block length $n$ and code rate $R_c \\triangleq\n\\frac{k}{n}$. In this case, the number of transmitted bits in $T_N$\nis increased from $N$-bit uncoded message to\n$\\frac{N}{R_c}=\\frac{N}{k}n$ bits coded one. To compute the energy\nconsumption of coded scheme, we use the fact that channel coding\nreduces the required average SNR value to achieve a given BER (i.e.,\nthe same BER as uncoded one). Taking this into account, the proposed\nWSN benefits in transmission energy saving of coded modulation\nspecified by\n$\\mathcal{E}_{t,c}=\\frac{\\mathcal{E}_{t}}{\\Upsilon_{c}}$, where\n$\\Upsilon_{c} \\geq 1$ is the coding gain\\footnote{Denoting\n$\\bar{\\gamma}\n=\\frac{\\Omega}{\\mathcal{L}_d}\\frac{\\mathcal{E}_t}{N_0}$ and\n$\\bar{\\gamma}_{c}\n=\\frac{\\Omega}{\\mathcal{L}_d}\\frac{\\mathcal{E}_{t,c}}{N_0}$ as the\naverage SNR of uncoded and coded schemes, respectively, the\n\\emph{coding gain} (expressed in dB) is defined as the difference\nbetween the values of $\\bar{\\gamma}$ and $\\bar{\\gamma}_{c}$ required\nto achieve a certain BER, where\n$\\mathcal{E}_{t,c}=\\frac{\\mathcal{E}_{t}}{\\Upsilon_{c}}$.} of the\nutilized coded MFSK. It should be noted that the cost of this energy\nsaving is the bandwidth expansion $\\frac{B}{R_{c}}$. In order to\nkeep the bandwidth of the coded system the same as that of the\nuncoded case, we must keep the information transmission rate\nconstant, i.e., the symbol duration $T_{s}$ of uncoded and coded\nMFSK would be the same. However, the active mode duration increases\nfrom $T_{ac}=\\frac{N}{b}T_{s}$ in the uncoded system to\n\\begin{equation}\\label{BCH_active}\nT_{ac,c}=\\frac{N}{bR_{c}}T_{s}=\\frac{T_{ac}}{R_{c}}\n\\end{equation}\nfor the coded case. Thus, one would assume that the total time $T_N$\nincreases to $\\frac{T_N}{R_c}$ for the coded scenario. It is worth\nmentioning that the active mode period in the coded case is upper\nbounded by $\\frac{T_N}{R_c}-T_{tr}$. As a result, the maximum\nconstellation size $M$, denoted by $M_{max}\\triangleq 2^{b_{max}}$,\nfor the coded MFSK is calculated by\n$\\frac{2^{b_{max}}}{b_{max}}=\\frac{B\nR_{c}}{N}(\\frac{T_{N}}{R_{c}}-T_{tr})$, which is approximately the\nsame as that of the uncoded case.\n\n\\newcounter{mytempeqncnt}\n\\begin{figure*}[!t]\n\\normalsize \\setcounter{mytempeqncnt}{\\value{equation}}\n\\setcounter{equation}{10}\n\\begin{eqnarray}\n\\left(\\left[1-\\left(1-\\frac{2(M-1)}{M}P_b\\right)^{\\frac{1}{M-1}}\\right]^{-1}-2\\right) \\dfrac{1}{\\log_2 M}&=& \\left(\\left[1-e^{\\frac{1}{M-1}\\ln(1-\\frac{2(M-1)}{M}P_b)}\\right]^{-1}-2\\right)\\dfrac{1}{\\log_2 M}\\\\\n&\\stackrel{(a)}{\\approx}& \\left(\\left[1-e^{-\\frac{2P_b}{M}}\\right]^{-1}-2\\right)\\dfrac{1}{\\log_2 M}\\\\\n\\label{mono10}&\\stackrel{(b)}{\\approx}&\\left(\\dfrac{M}{2P_b}-2\\right)\\dfrac{1}{\\log_2\nM},\n\\end{eqnarray}\n\\setcounter{equation}{\\value{mytempeqncnt}} \\hrulefill \\vspace*{4pt}\n\\end{figure*}\n\nWe denote $\\mathcal{E}_{enc}$ and $\\mathcal{E}_{dec}$ as the\ncomputation energy of the encoder and decoder for each information\nbit, respectively. Thus, the total computation energy cost of the\ncoding components for $\\frac{N}{R_{c}}$ bits is obtained as\n$N\\frac{\\mathcal{E}_{enc}+\\mathcal{E}_{dec}}{R_{c}}$. Substituting\n(\\ref{deriv1}) in\n$\\mathcal{E}_{t,c}=\\frac{\\mathcal{E}_{t}}{\\Upsilon_{c}}$, and using\n(\\ref{sensor_power}), (\\ref{sink_power}) and (\\ref{BCH_active}), the\ntotal energy consumption of transmitting $\\frac{N}{R_c}$ bits in\neach period $\\frac{T_N}{R_C}$ for an arbitrarily coded MFSK, and a\ngiven $P_b$ is obtained as\n\\begin{eqnarray}\n\\notag \\mathcal{E}_{N,c} &=& (1+\\alpha) \\left[\\left(\n1-\\left(1-\\dfrac{2(M-1)}{M}P_b\\right)^{\\frac{1}{M-1}}\\right)^{-1}-2\n\\right]\\\\\n\\notag&\\times&\\dfrac{\\mathcal{L}_d N_0}{\\Omega \\Upsilon_{c}}\n\\dfrac{N}{R_{c}\\log_2 M}+\n(\\mathcal{P}_{c}-\\mathcal{P}_{Amp})\\dfrac{MN}{BR_{c} \\log_{2}M}+ \\\\\n\\label{energy_totalcodedFSK} && 2\n\\mathcal{P}_{Sy}T_{tr}+N\\frac{\\mathcal{E}_{enc}+\\mathcal{E}_{dec}}{R_{c}}.\n\\end{eqnarray}\nTo make a fair comparison between the uncoded and coded modulation,\nwe use the same BER and drop the subscript ``$c$'' for $P_b$ in the\ncoded case. Thus, the optimization goal is to minimize the objective\nfunction $\\mathcal{E}_{N,c}$ over modulation and coding parameters.\nThis is achieved by finding the optimum constellation size $M$ under\nthe constraint $2\\leq M \\leq M_{max}$ for a specific channel coding\nscheme, and then minimize $\\mathcal{E}_{N,c}$ over the coding\nparameters.\n\nFor the above optimization problem, we consider two scenarios: $i)$\nfixed-rate codes (e.g., BCH and convolutional codes), and $ii)$\nvariable rate codes (e.g., LT codes). To find the optimum $M$ for a\ngiven fixed-rate code, we prove that (\\ref{energy_totalcodedFSK}) is\na monotonically increasing function of $M$ for every value of $d$\nand $\\eta$. Since $B$ and $N$ are fixed and $R_c$ is independent of\n$M$, it is concluded that the second term in\n(\\ref{energy_totalcodedFSK}) is a monotonically increasing function\nof $M$. Also, from the first term in (\\ref{energy_totalcodedFSK}),\nwe have (11)-(13) in the top of this page,\nwhere $(a)$ comes from the approximation $\\ln(1-z)\\approx -z,~~|z|\n\\ll 1$, and the fact that $P_b$ scales as $o(1)$. Also, $(b)$\nfollows from the approximation\n$e^{-z}=\\sum_{n=0}^{\\infty}(-1)^{n}\\frac{z^n}{n!}\\approx\n1-z,~~|z|\\ll1$. On the other hand, it is shown in \\cite{Proakis2001}\nthat $\\Upsilon_c$ for MFSK is a decreasing function of $M$. Thus, it\nis concluded from (\\ref{mono10}) that the first term in\n(\\ref{energy_totalcodedFSK}) is also a monotonically increasing\nfunction of $M$. As a result, the minimum total energy consumption\n$\\mathcal{E}_{N,c}$ for a given fixed-rate code is achieved at\n$\\hat{M}=2$.\n\nIn the next section, we evaluate the above optimization problem for\nthe LT codes using some simulation studies on the probability mass\nfunction of the LT code rate and coding gain. We show that the LT\ncode parameters depend strongly on the constellation size $M$ and\nexhibit different trends over fixed-rate codes. In addition, we\npresent some beneficial uses of LT codes over block and\nconvolutional codes in managing the energy consumption for different\nchannel realizations.\n\n\n\n\n\\subsection{Energy Optimality of LT Codes}\nLT codes are the first class of Fountain codes which usually\nspecified by the number of input bits $k$ and the output-node degree\ndistribution $\\mathcal{O}(x)$. Without loss of generality and for\nease of our analysis, we assume that a single $k$-bit message\n$\\mathcal{B}_1 \\triangleq (m_1,m_2,...,m_k)\\in \\mathcal{M}_N$ is\nencoded to codeword $\\mathcal{C}_1 \\triangleq (a_1,a_2,...,a_n)$.\nEach single coded bit $a_i$ is generated based on the encoding\nprotocol proposed in \\cite{LubyFOCS2002}: $i)$ randomly choose a\ndegree $1 \\leq \\mathcal{D} \\leq k$ from $\\mathcal{O}(x)$, $ii)$\nusing a uniform distribution, randomly choose $\\mathcal{D}$ distinct\ninput bits, and calculate the encoded bit $a_i$ as the XOR-sum of\nthese $\\mathcal{D}$ bits. The above encoding process defines a\n\\emph{bipartite graph} connecting encoded nodes to input nodes (see,\ne.g., Fig. \\ref{fig: LT Encoder}). It is seen that the LT encoding\nprocess is extremely simple and has very low energy consumption.\nUnlike block and convolutional codes, in which the codeword block\nlength is fixed, for the above LT code, $n$ is a variable parameter,\nresulting in a random variable LT code rate $R^{LT}_c \\triangleq\n\\frac{k}{n}$. More precisely, $a_n \\in \\mathcal{C}_1$ is the last\nbit generated at the output of LT encoder before receiving the\nacknowledgement signal from the sink node indicating termination of\na successful decoding process. This inherent property of LT codes\nmeans they can vary their codeword block lengths to adapt to any\nwireless channel condition between the sensor and the sink nodes.\n\n\\begin{figure}[t]\n\\centerline{\\psfig{figure=LT_Code1.eps,width=3.25in}} \\caption{\nBipartite graph of an LT code with $k=6$ and $n=8$. } \\label{fig: LT\nEncoder}\n\\end{figure}\n\n\nTo describe the output-node degree distribution used in this work,\nlet $\\mu_i$, $i=1,...,k$, denote the probability that an output node\nhas degree $i$. Following the notation of\n\\cite{Shokrollahi_ITIT0606}, the output-node degree distribution of\nan LT code has the polynomial form $\\mathcal{O}(x) \\triangleq\n\\sum_{i=1}^{k}\\mu_i x^i$ with the property that $\\mathcal{O}(1) =\n\\sum_{i=1}^{k}\\mu_i =1$. Typically, optimizing the output-node\ndegree distribution for a specific wireless channel model is a\ncrucial task in designing LT codes. In fact, for wireless fading\nchannels, it is still an open problem, what the ``optimal\"\n$\\mathcal{O}(x)$ is. In this work, we use the following output-node\ndegree distribution which was optimized for a BSC using a\nhard-decision decoder \\cite{David_Thesis2008}:\n\\begin{eqnarray}\n\\notag \\mathcal{O}(x)&=&0.00466x+0.55545x^2+0.09743x^3+\\\\\n\\notag &&0.17506x^5+0.03774x^8+0.08202x^{14}+\\\\\n\\label{degree}&&0.01775x^{33}+0.02989x^{100}.\n\\end{eqnarray}\n\nThe LT decoder at the sink node can recover the original $k$-bit\nmessage $\\mathcal{B}_1$ with high probability after receiving any\n$(1+\\epsilon)k$ bits in its buffer, where $\\epsilon$ depends upon\nthe LT code design \\cite{Shokrollahi_ITIT0606}. For this recovery\nprocess, the LT decoder needs to correctly reconstruct the bipartite\ngraph of an LT code. One practical approach suitable for the\nproposed WSN model is that the LT encoder and decoder use identical\npseudo-random generators with a common seed value which may reduce\nthe complexity further. In this work, we assume that the sink node\nrecovers $k$-bit message $\\mathcal{B}_1$ using a simple\nhard-decision ``\\emph{ternary message passing}\" decoder in a nearly\nidentical manner to the ``\\emph{Algorithm E}'' decoder in\n\\cite{RichardsonITIT0201} for Low-Density Parity-Check (LDPC)\ncodes\\footnote{Description of the \\emph{ternary message passing}\ndecoding is out of scope of this work, and the reader is referred to\nChapter 4 in \\cite{David_Thesis2008} for more details.}. Also, the\ndegree distribution $\\mathcal{O}(x)$ in (\\ref{degree}) was optimized\nfor a ternary decoder in a BSC and we are aware of no better\n$\\mathcal{O}(x)$ for the ternary decoder in Rayleigh fading\nchannels.\n\nUnlike fixed-rate codes in which the active mode duration of coded\nMFSK is fixed, for the LT coded MFSK, we have a non-fixed value for\n\\begin{equation}\\label{active_LT}\nT_{ac,c}^{LT}=\\dfrac{N}{b R_{c}^{LT}}T_s=\\dfrac{MN}{BR^{LT}_{c}\n\\log_{2}M}.\n\\end{equation}\nAn interesting point raised from (\\ref{active_LT}) is that\n$T_{ac,c}^{LT}$ is a function of the random variable $R_c^{LT}$\nwhich results in an inherent adaptive duty-cycling for power\nmanagement in each channel condition without any channel state\ninformation fed back from the sink node to the sensor node.\nRecalling from (\\ref{energy_totalcodedFSK}), we have\n\\begin{eqnarray}\n\\notag \\mathcal{E}^{LT}_{N,c} &=& (1+\\alpha) \\left[\\left(\n1-\\left(1-\\dfrac{2(M-1)}{M}P_b\\right)^{\\frac{1}{M-1}}\\right)^{-1}-2\n\\right]\\\\\n\\notag &\\times&\\dfrac{\\mathcal{L}_d N_0}{\\Omega \\Upsilon^{LT}_{c}}\n\\dfrac{N}{R^{LT}_{c}\\log_2 M}+\n(\\mathcal{P}_{c}-\\mathcal{P}_{Amp})\\dfrac{MN}{BR^{LT}_{c}\n\\log_{2}M}+ \\\\\n\\notag&&2\n\\mathcal{P}_{Sy}T_{tr}+N\\frac{\\mathcal{E}^{LT}_{enc}+\\mathcal{E}^{LT}_{dec}}{R^{LT}_{c}}.\n\\end{eqnarray}\nwhere the main goal is to minimize $\\mathcal{E}^{LT}_{N,c}$ over $M$\nand the coding parameters. Toward this goal, we first compute the LT\ncode rate and the corresponding LT coding gain. Let us begin with\nthe case of asymptotic LT code rate, where the number of input bits\n$k$ goes to infinity. It is shown that the LT code rate is obtained\nasymptotically as a fixed value of $R^{LT}_c \\approx\n\\frac{\\mathcal{O}_{ave}}{\\mathcal{I}_{ave}}$ for large values of\n$k$, where $\\mathcal{O}_{ave}$ and $\\mathcal{I}_{ave}$ represent the\naverage degree of the output and the input nodes in the bipartite\ngraph, respectively (see Appendix I for the proof).\n\n\n\nWhen using finite-$k$ LT codes, we are treating the channel as\nstatic over one block length. In this case, the rate of the LT code\nfor any block can be chosen to achieve the desired performance for\nthat block. More precisely, for the particular block we are\nconcerned with, the receiver could evaluate the channel\ninstantaneous SNR, and then determine how many code bits it needs to\ncollect in order to achieve its given BER target, thus essentially\ndynamically selecting its rate. For the next block, the receiver\nwould once again evaluate the (new) instantaneous channel SNR and\nadjust its rate accordingly to collect more or fewer bits.\n\nIt should be noted that there is no (currently known) explicit equation\ngoverning the relationship between the instantaneous SNR and the required\nnumber of decoding bits for an LT code. In this work, we determine\nthe necessary number of decoding bits in each case through simulation,\nas described below:\n\\begin{itemize}\n\\item [$\\textbf{i)}$] We decide upon a ``\\emph{target BER}'' for the decoded bits - e.g., the\ndecoded message needed an average BER of $10^{-4}$ or better.\n\\item [$\\textbf{ii)}$] Based on the assumption that instantaneous SNR is constant over at\nleast one block we perform the following steps for a large range of possible\ninstantaneous SNR:\n\n- Determine through numerical simulation the decoded BER using the LT code rate $R^{LT}_c= 1$.\n\n- If the decoded BER is greater than the target BER, then we reduce the LT code rate $R^{LT}_c$ and try again.\n\n- Repeat the above step until the decoded BER is less than the target BER.\n\n\\item [$\\textbf{iii)}$] At this point, for the given LT degree distribution, decoder, and SNR\nwe can identify the highest LT code rate that will yield the target BER\nor better.\n\\end{itemize}\nArmed with this information (computed ahead of time), a receiver can determine\nthe appropriate rate at which to operate and hence determine how many bits it\nmust collect for a given instantaneous SNR.\nThis is also based on the common assumption that the receiver is capable of\nestimating the instantaneous SNR.\n\n\n\nBased on the above arguments and for any given average SNR, the LT code rate is described\nby either a probability mass function (pmf) or a probability density\nfunction (pdf) denoted by $P_R(\\ell)$. Because it is difficult to\nget a closed-form expression of $P_R(\\ell)$, we use a discretized\nnumerical method to calculate the pmf $P_R(\\ell)\\triangleq\n\\textrm{Pr}\\{R_c^{LT}=\\ell \\}$, $0 \\leq \\ell \\leq 1$, for different\nvalues of $M$.\nWe plot the pmfs of the LT code rate in Fig. \\ref{fig: LT code_rate_mpf}\nfor $M=2$ and various average SNR \\footnote{It should be noted that\nthe average SNR is expressed in terms of $\\frac{\\mathcal{E}_b}{N_0}$,\nwhere $\\mathcal{E}_b$ represents the transmit energy per bit.} over a Rayleigh fading channel model.\nIt is observed that for lower average SNRs the pmfs are larger in\nthe lower rate regimes (i.e., the pmfs spend more time in the low\nrate region). Also, all pmfs exhibit quite a spike for the highest\nrate, which makes sense since once the instantaneous SNR hits a\ncertain critical value, the codes will always decode with a high\nrate. Also, Fig. \\ref{fig: LT code_rate_pmf_M} illustrates the pmf\nof LT code rates for various constellation size $M$ and for average\nSNR equal to 16 dB. It can be seen that as $M$ increases, the rate\nof the LT code tends to have a pmf with larger values in the lower\nrate regions.\n\n\\begin{figure}[t]\n\\centerline{\\psfig{figure=LT_Code_Rate_pmf.eps,width=3.75in}}\n\\caption{The pmf of LT code rate for various average SNR and M=2. }\n\\label{fig: LT code_rate_mpf}\n\\end{figure}\n\n\n\n\\begin{figure}[t]\n\\centerline{\\psfig{figure=LT_Code_Rate_pmf_M.eps,width=3.75in}}\n\\caption{The pmf of LT code rate for various constellation size $M$\nand average SNR=16 dB. } \\label{fig: LT code_rate_pmf_M}\n\\end{figure}\n\n\nTable II illustrates the average LT code rates and the\ncorresponding coding gains of LT coded MFSK using $\\mathcal{O}(x)$\nin (\\ref{degree}), for $M=2,4,8,16$ and given $P_b=10^{-3}$. The\naverage rate for a certain average SNR is obtained by integrating\nthe pmf over the rates from $0$ to $1$. To get more insight into\nthe relationship between the LT code rate and coding gain, we plot\nthe results of Table II in Fig. \\ref{fig: LT code_rate_coding_gain}. It is observed that the LT\ncode is able to provide a huge coding gain $\\Upsilon_{c}^{LT}$ given\n$P_b=10^{-3}$, but this gain comes at the expense of a very low\naverage code rate, which means many additional code bits need to be\nsent. This results in higher energy consumption per information bit.\nIn contrast to the fixed-rate codes in which the coding gain\ndecreases when $M$ grows, the LT coding gains display different\ntrends in terms of $M$ as illustrated in Table II. Thus, in\ncontrast with fixed-rate codes, $\\mathcal{E}_{N,c}^{LT}$ is not\nnecessarily a monotonically increasing function of $M$. In the next\nsection, we evaluate numerically $\\mathcal{E}_{N,c}^{LT}$ in terms\nof the optimized modulation and coding parameters compared to the\nuncoded and the fixed-rate codes.\n\n\\begin{figure*}[t]\n\\centerline{\\psfig{figure=LT_Codig_Gain_vs_Rate.eps,width=6.15in}}\n\\caption{LT coding gain versus average LT code rate for $P_b=10^{-3}$ and\nM=2,4,8,16. } \\label{fig: LT code_rate_coding_gain}\n\\end{figure*}\n\n\nAn interesting point extracted from Table II is the flexibility of\nthe LT code to adjust its rate (and its corresponding coding gain)\nto suit instantaneous channel conditions in WSNs. For instance in\nthe case of favorable channel conditions, the LT coded MFSK is able\nto achieve $R_{c}^{LT}\\approx1$ with $\\Upsilon_{c}^{LT}\\approx 0$\ndB, which is similar to the case of uncoded MFSK, i.e., $n=k$. The\neffect of LT code rate flexibility on the total energy consumption\nis also observed in the simulation results in the subsequent\nsection.\n\n\n\\begin{table*}\n\\label{table0021} \\caption{Average LT code rate and Coding gain of\nLT coded MFSK over Rayleigh fading model for $P_b=10^{-3}$ and\nM=2,4,8,16.} \\centering\n \\begin{tabular}{|c|cc||c|cc|cc|cc|}\n \\hline\n & M=2 & & & M=4 & & M=8 & & M=16 & \\\\\n \\hline\n $\\frac{\\mathcal{E}_b}{N_0}$ & Average & Coding & $\\frac{\\mathcal{E}_b}{N_0}$ & Average & Coding & Average & Coding & Average & Coding \\\\\n (dB) & Code Rate & Gain (dB) & (dB) & Code Rate & Gain (dB) & Code Rate & Gain (dB) & Code Rate & Gain (dB) \\\\\n \\hline\n 5 & 0.2560 & 25 & 0 & 0.0028 & 33.87 & 0.0012 & 36.46 & 0.0012 & 38.48 \\\\\n 6 & 0.3174 & 24 & 2 & 0.0140 & 31.87 & 0.0024 & 34.46 & 0.0012 & 36.48 \\\\\n 7 & 0.3819 & 23 & 4 & 0.0460 & 29.87 & 0.0095 & 32.46 & 0.0021 & 34.48 \\\\\n 8 & 0.4475 & 22 & 6 & 0.1100 & 27.87 & 0.0330 & 30.46 & 0.0086 & 32.48 \\\\\n 9 & 0.5120 & 21 & 8 & 0.2100 & 25.87 & 0.0870 & 28.46 & 0.0320 & 30.48 \\\\\n 10& 0.5738 & 20 & 10 & 0.3300 & 23.87 & 0.1800 & 26.46 & 0.0870 & 28.48 \\\\\n 11& 0.6315 & 19 & 12 & 0.4600 & 21.87 & 0.3000 & 24.46 & 0.1800 & 26.48 \\\\\n 12& 0.6840 & 18 & 14 & 0.5900 & 19.87 & 0.4400 & 22.46 & 0.3100 & 24.48 \\\\\n 13& 0.7307 & 17 & 16 & 0.7000 & 17.87 & 0.5700 & 20.46 & 0.4500 & 22.48 \\\\\n 14& 0.7716 & 16 & 18 & 0.7800 & 15.87 & 0.6800 & 18.46 & 0.5800 & 20.48 \\\\\n 15& 0.8067 & 15 & 20 & 0.8500 & 13.87 & 0.7700 & 16.46 & 0.6900 & 18.48 \\\\\n 16& 0.8365 & 14 & 22 & 0.8900 & 11.87 & 0.8400 & 14.46 & 0.7800 & 16.48 \\\\\n 17& 0.8614 & 13 & 24 & 0.9200 & 9.87 & 0.8800 & 12.46 & 0.8400 & 14.48 \\\\\n 18& 0.8821 & 12 & 26 & 0.9400 & 7.87 & 0.9100 & 10.46 & 0.8800 & 12.48 \\\\\n 19& 0.8991 & 11 & 28 & 0.9500 & 5.87 & 0.9300 & 8.46 & 0.9200 & 10.48 \\\\\n 20& 0.9130 & 10 & 30 & 0.9600 & 3.87 & 0.9500 & 6.46 & 0.9400 & 8.48 \\\\\n 22& 0.9333 & 8 & 32 & 0.9600 & 1.87 & 0.9500 & 4.46 & 0.9500 & 6.48 \\\\\n 24& 0.9466 & 6 & 34 & 0.9600 & -0.13 & 0.9600 & 2.46 & 0.9500 & 4.48 \\\\\n 26& 0.9551 & 4 & 36 & 0.9700 & -2.13 & 0.9600 & 0.46 & 0.9600 & 2.48 \\\\\n 28& 0.9606 & 2 & 38 & 0.9700 & -4.13 & 0.9700 & -1.54 & 0.9600 & 0.48 \\\\\n 30& 0.9640 & 0 & 40 & 0.9700 & -6.13 & 0.9700 & -3.54 & 0.9700 & -1.52 \\\\\n\n \\hline\n \\end{tabular}\n\\end{table*}\n\n\n\n\n\\section{Numerical Evaluations}\\label{simulation_Ch5}\nIn this section, we present some numerical evaluations using\nrealistic parameters from the IEEE 802.15.4 standard and\nstate-of-the art technology to confirm the energy efficiency\nanalysis of uncoded and coded MFSK modulations discussed in Sections\n\\ref{uncoded_MFSK} and \\ref{analysis_Ch4}.\n\n\\subsection{Experimental Setup}\nWe assume that MFSK operates in the carrier frequency $f_0=$2.4 GHz\nIndustrial Scientist and Medical (ISM) unlicensed band utilized in\nthe IEEE 802.15.14 standard \\cite{IEEE_802_15_4_2006}. According to\nthe FCC 15.247 RSS-210 standard for United States\/Canada, the\nmaximum allowed antenna gain is 6 dBi \\cite{FreeScale2007}. In this\nwork, we assume that $\\mathcal{G}_t=\\mathcal{G}_r=5$ dBi. Thus for\nthe $f_0=$2.4 GHz, $\\mathcal{L}_1~ \\textrm{(dB)}\\triangleq\n10\\log_{10}\\left(\\frac{(4 \\pi)^2}{\\mathcal{G}_t \\mathcal{G}_r\n\\lambda^2}\\right) \\approx 30~ \\textrm{dB}$, where $\\lambda\n\\triangleq \\frac{3\\times 10^8}{f_0}=0.125$ m. We assume that in each\nperiod $T_N$, the data frame $N=1024$ bytes (or equivalently\n$N=8192$ bits) is generated, where $T_N$ is assumed to be 1.4 s. The\nchannel bandwidth is set to the $B=62.5$ KHz, according to IEEE\n802.15.4 \\cite[p. 49]{IEEE_802_15_4_2006}. It is concluded from\n$\\frac{2^{b_{max}}}{b_{max}}=\\frac{BR_c}{N}(\\frac{T_{N}}{R_c}-T_{tr})\\approx\n\\frac{B}{N}T_N$ that $M_{max} \\approx 64$ (or equivalently $b_{max}\n\\approx 6$) for MFSK. Table III summarizes the system parameters for\nsimulation\\footnote{To make a fair comparison between the energy\nconsumption of different communication\nschemes, the bandwidth and the BER are assumed to be the same for all the schemes.} \\cite{Cui_GoldsmithITWC0905, TangITWC0407,\nBevilacqua_IJSSC1204, Wang_ISLPED2001}. The results in Tables II\nare also used to compare the energy efficiency of uncoded and\ncoded MFSK schemes.\n\n\\begin{table}\n\\label{table001} \\caption{System Evaluation Parameters} \\centering\n \\begin{tabular}{lll}\n \\hline\n $B=62.5$ KHz & $N_0=-180$ dB & $\\mathcal{P}_{ADC}=7$ mw \\\\\n\n $M_l=40$ dB & $\\mathcal{P}_{Sy}=10$ mw & $\\mathcal{P}_{LNA}=9$ mw\\\\\n\n $\\mathcal{L}_1=30$ dB & $\\mathcal{P}_{Filt}=2.5$ mw & $\\mathcal{P}_{ED}=3$ mw \\\\\n\n $\\eta=3.5$ & $\\mathcal{P}_{Filr}=2.5$ mw & $\\mathcal{P}_{IFA}=3$ mw \\\\\n\n $\\Omega=1$ & $T_N=1.4$ sec & $T_{tr}=5~\\mu s$ \\\\\n\n \\hline\n \\end{tabular}\n\\end{table}\n\n\nIn order to estimate the computation energy of a specific channel\ncoding, we use the ARM7TDMI core which is the industry's most widely\nused 32-bit embedded RISC microprocessor for an accurate power\nsimulation \\cite{ARMTech2004}. For the energy and power\ncalculations, the relations\n$\\mathcal{E}_{operation}=\\dfrac{n_c}{n_o}\\times \\mathcal{E}_{Hz}$\nand $\\mathcal{E}_{block}=n_o \\times \\mathcal{E}_{operation}$ are\nused, where\n\\begin{itemize}\n\\item [] $n_o:$ number of operations per block.\n\\item [] $n_c:$ number of clock cycles on $n_o$ operations.\n\\item [] $\\mathcal{E}_{Hz}:$ total dynamic power consumption per\nHertz (W\/Hz) of a single calculation cycle.\n\\end{itemize}\nIn addition, for the purpose of comparative evaluation, we use some\nclassical BCH$(n,k,t)$ codes with $t$-error correction capability,\nand convolutional codes $(n,k,L)$ with the constraint length $L$.\nThese codes are widely utilized in IEEE standards \\cite{IEEE802.15,\nIEEE802.16}. We use hard-decision prior to decoding for the BCH and\nconvolutional codes. The main reason for using the hard-decision\nhere is to make a fair comparison to the LT code, since this code\ninvolved a hard-decision in ternary decoder.\n\n\n\\subsection{Optimal Configuration}\nAs a starting point, we obtain the coding gain of some practical BCH\nand convolutional codes with MFSK for different constellation size\n$M$ and given $P_b=(10^{-3},10^{-4})$ in Table IV. We observe from\nTable IV that there is a tradeoff between coding gain and decoder\ncomplexity for the BCH codes. In fact, achieving a higher coding\ngain for a given $M$, requires a more complex decoding process,\n(i.e., higher $t$) with more circuit power consumption. In addition,\nit is seen from Fig. \\ref{fig: BCH code_rate_coding_gain} that the coding gain of the BCH code is a monotonically\ndecreasing function of $M$ as expected. For a given $M$, the\nconvolutional codes with lower rates and higher constraint lengths\nachieve greater coding gains. In contrast to\n\\cite{Cui_GoldsmithITWC0905}, where the authors assume a fixed\nconvolutional coding gain for every value of $M$, it is observed\nthat the coding gain of convolutional coded MFSK is a monotonically\ndecreasing function of $M$. By comparing the results in Table II\nwith those in Table IV for BCH and convolutional codes, one observes\nthat LT codes outperform the other coding schemes in energy saving\nat comparable rates.\n\n\\begin{figure}[t]\n\\centerline{\\psfig{figure=BCH_Coding_Gain_vs_Code_Rate.eps,width=4.15in}}\n\\caption{Coding gain versus code rate of some practical BCH$(n,k,t)$ codes for $P_b=10^{-3}$ and\nM=2,4,8,16. } \\label{fig: BCH code_rate_coding_gain}\n\\end{figure}\n\n\n\\begin{table*}\n\\label{table01} \\caption{Coding gain ($dB$) of BCH and convolutional\ncoded MFSK over Rayleigh fading for BER$=(10^{-3},10^{-4})$.}\n\\centering\n \\begin{tabular}{|l|c| c c c c c c|}\n \\hline\n BCH Code$(n,k,t)$ & $R^{BC}_c$ & M=2 & M=4 & M=8 & M=16 & M=32 & M=64\n \\\\\n \\hline\n BCH $(7,4,1)$ & 0.571 & $(2.5,2.8)$ & $(0.3,0.4)$ & $(0.1,0.2)$ & $(0.0,0.0)$ & $(0.0,0.0)$ & $(0.0,0.0)$ \\\\\n \\hline\n BCH $(15,11,1)$ & 0.733 & $(1.4,1.6)$ & $(0.2,0.3)$ & $(0.0,0.0)$ & $(0.0,0.0)$ & $(0.0,0.0)$ & $(0.0,0.0)$ \\\\\n \\hline\n BCH $(15,7,2)$ & 0.467 & $(2.4,3.3)$ & $(2.0,2.3)$ & $(0.8,1.0)$ & $(0.3,0.4)$ & $(0.0,0.0)$ & $(0.0,0.0)$ \\\\\n \\hline\n BCH $(15,5,3)$ & 0.333 & $(4.1,4.6)$ & $(2.7,2.9)$ & $(2.0,2.1)$ & $(1.5,1.60)$ & $(0.7,0.8)$ & $(0.2,0.2)$ \\\\\n \\hline\n BCH $(31,26,1)$ & 0.839 & $(1.2,1.5)$ & $(0.2,0.2)$ & $(0.0,0.0)$ & $(0.0,0.0)$ & $(0.0,0.0)$ & $(0.0,0.0)$ \\\\\n \\hline\n BCH $(31,21,2)$ & 0.677 & $(2.3,2.9)$ & $(1.7,2.0)$ & $(0.7,0.8)$ & $(0.2,0.2)$ & $(0.0,0.0)$ & $(0.0,0.0)$ \\\\\n \\hline\n BCH $(31,16,3)$ & 0.516 & $(2.9,3.1)$ & $(2.1,2.2)$ & $(1.5,1.6)$ & $(1.3,1.4)$ & $(0.6,0.7)$ & $(0.1,0.1)$ \\\\\n \\hline\n BCH $(31,11,5)$ & 0.355 & $(4.1,4.4)$ & $(3.5,4.2)$ & $(2.2,2.3)$ & $(2.0,2.1)$ & $(1.8,2.0)$ & $(1.1,1.3)$ \\\\\n \\hline\n BCH $(31,6,7)$ & 0.194 & $(5.4,5.9)$ & $(4.3,4.8)$ & $(3.5,3.8)$ & $(3.2,3.3)$ & $(2.7,2.8)$ & $(2.3,2.4)$ \\\\\n \\hline\n \\hline\n Convolutional Code & $R^{CC}_c$ & M=2 & M=4 & M=8 & M=16 & M=32 & M=64\\\\\n \\hline\n trel$(6,[53~~75])$ & 0.500 & $(3.8,4.6)$ & $(2.7,3.1)$ & $(2.1,2.3)$ & $(1.8,2.0)$ & $(1.4,1.5)$ & $(1.4,1.4)$ \\\\\n \\hline\n trel$(7,[133~~171])$ & 0.500 & $(4.0,4.7)$ & $(3.0,3.5)$ & $(2.2,2.4)$ & $(1.8,2.0)$ & $(1.5,1.6)$ & $(1.4,1.5)$ \\\\\n \\hline\n trel$(7,[133~~165~~171])$ & 0.333 & $(5.7,6.4)$ & $(4.8,5.1)$ & $(3.7,3.9)$ & $(3.1,3.3)$ & $(2.7,2.8)$ & $(2.5,2.6)$ \\\\\n \\hline\n trel$([4~3],[4~5~17;7~4~2])$ & 0.667 & $(2.2,2.6)$ & $(1.5,1.7)$ & $(0.9,1.1)$ & $(0.6,0.6)$ & $(0.5,0.5)$ & $(0.5,0.5)$ \\\\\n \\hline\n trel$([5~4],[23~35~0;0~5~13])$ & 0.667 & $(2.9,3.5)$ & $(1.9,2.4)$ & $(1.4,1.8)$ & $(1.1,1.2)$ & $(0.8,0.9)$ & $(0.7,0.8)$ \\\\\n \\hline\n \\end{tabular}\n\\end{table*}\n\n\n\\begin{figure}[t]\n\\centerline{\\psfig{figure=Coded_Uncoded.eps,width=3.9in}}\n\\caption{Total energy consumption of optimized coded and uncoded\nMFSK versus $d$ for $P_b=10^{-3}$.} \\label{fig: Coded_Uncoded}\n\\end{figure}\n\nFig. \\ref{fig: Coded_Uncoded} shows the total energy consumption\nversus distance $d$ for the optimized BCH, convolutional and LT\ncoded MFSK schemes, compared to the optimized uncoded MFSK for\n$P_b=10^{-3}$. The optimization is done over $M$ and the parameters\nof coding scheme. Simulation results show that for $d$ less than the\nthreshold level $d_T \\approx 40$ m, the total energy consumption of\noptimized uncoded MFSK is less than that of the coded MFSK schemes.\nHowever, the energy gap between LT coded and uncoded MFSK is\nnegligible compared to the other coded schemes as expected. For\n$d>d_T$, the LT coded MFSK scheme is more energy-efficient than\nuncoded and other coded MFSK schemes. Also, it is observed that the\nenergy gap between LT and convolutional coded MFSK increases when\nthe distance $d$ grows. This result comes from the high coding gain\ncapability of LT codes which confirms our analysis in Section\n\\ref{analysis_Ch4}. The threshold level $d_T$ (for LT code) or\n$d^{'}_T$ (for BCH and convolutional codes) are obtained when the\ntotal energy consumptions of coded and uncoded systems become equal.\nFor instance, using $\\mathcal{L}_d =M_ld^\\eta \\mathcal{L}_1$, and\nthe equality between (\\ref{energy_totFSK}) and\n(\\ref{energy_totalcodedFSK}) for uncoded and convolutional coded\nMFSK, we have\n\\begin{equation}\nd^{'}_T=\\left[\\hat{\\Upsilon}_c^{CC}\\dfrac{a_2\n(1-\\hat{R}_c^{CC})+N(\\mathcal{E}^{CC}_{enc}+\\mathcal{E}^{CC}_{dec})}{a_1(\\hat{\\Upsilon}_{c}^{CC}\\hat{R}_c^{CC}-1)}\\right]^{\\frac{1}{\\eta}},\n\\end{equation}\nwhere\n\\begin{eqnarray}\n\\notag a_1 &\\triangleq &(1+\\alpha) \\left[\\left(\n1-\\left(1-\\dfrac{2(M-1)}{M}P_b\\right)^{\\frac{1}{M-1}}\\right)^{-1}-2\n\\right]\\\\\n&\\times&\\dfrac{\\mathcal{L}_1 M_l}{\\log_2 M}\\dfrac{N N_0}{\\Omega},\n\\end{eqnarray}\n\\begin{equation}\na_2 \\triangleq (\\mathcal{P}_{c}-\\mathcal{P}_{Amp})\\dfrac{MN}{B\n\\log_2 M},\n\\end{equation}\n\\begin{equation}\n(\\hat{\\Upsilon}_c^{CC},\\hat{R}_c^{CC})=\n\\textrm{arg}~\\min_{\\Upsilon_c^{CC},R_c^{CC}}~\\mathcal{E}^{CC}_{N,c}.\n\\end{equation}\n\n It should be noted that the above threshold level imposes a\nconstraint on the design of the physical layer of some wireless\nsensor networking applications, in particular dynamic WSNs. To\nobtain more insight into this issue, let assume that the location of\nthe sensor node is changed every $T_d \\gg T_c$ time unit, where\n$T_c$ is the channel coherence time. For the moment, let us assume\nthat the sensor node aims to choose either a \\emph{fixed-rate coded}\nor an uncoded MFSK based on the distance between sensor and sink\nnodes. According to the results in Fig. \\ref{fig: Coded_Uncoded}, it\nis revealed that using fixed-rate channel coding is not energy\nefficient for short distance transmission (i.e., $d < d^{'}_T$),\nwhile for $d > d^{'}_T$, convolutional coded MFSK is more\nenergy-efficient than other schemes. For this configuration, the\nsensor node must have the capability of an adaptive coding scheme\nfor each distance $d$. However, as discussed previously, the LT\ncodes can adjust their rates for each channel condition and have\n(with a good approximation) minimum energy consumption for every\ndistance $d$. This indicates that LT codes can surpass the above\ndistance constraint for WSN applications with dynamic position\nsensor nodes over Rayleigh fading channels. This characteristic of\nLT codes results in reducing the complexity of the network design as\nwell. Of interest is the strong benefits of using LT coded MFSK\ncompared with the coded modulation schemes in\n\\cite{Cui_GoldsmithITWC0905, Chouhan_ITWC1009}. In contrast to\nclassical fixed-rate codes used in \\cite{Cui_GoldsmithITWC0905,\nChouhan_ITWC1009}, the LT codes can vary their block lengths to\nadapt to any channel condition in each distance $d$. Unlike\n\\cite{Cui_GoldsmithITWC0905} and \\cite{Chouhan_ITWC1009}, where the\nauthors consider fixed-rate codes over an AWGN channel model, we\nconsidered a Rayleigh fading channel which is a general model in\npractical WSNs. The simplicity and flexibility advantages of LT\ncodes with an MFSK scheme make them the preferable choice for\nwireless sensor networks, in particular for WSNs with dynamic\nposition sensor nodes.\n\n\\textbf{Remark 1:} As discussed previously, the proposed LT coded scheme\nbenefits in adjusting the coding parameters in each channel realization\nor equivalently each distance $d$ to minimize the total energy consumption.\nAs we will show shortly, this comes from a variable transmission time process\nwhich adaptively controls the power consumption of the proposed scheme.\nTo address this issue, we plot the active mode duration of the optimized\nLT coded modulation versus $d$ compared to those of the optimized uncoded\nand other coded modulation schemes in Fig. \\ref{fig: active}.\nFor this purpose, we use $T_{ac,c}=\\frac{T_{ac}}{R_c}$, where $T_{ac}=\\frac{MN}{B \\log_2 M}$\ndenotes the active mode duration of uncoded scheme. Simulation results show\nthat the optimum constellation size $M$ that minimizes the total energy\nconsumption for the aforementioned schemes and for every value of distance\n$d$ is $\\hat{M}=2$. It is revealed from Fig. \\ref{fig: active} that the\noptimized uncoded and convolutional coded MFSK display fixed values of\n$T_{ac}$ and $T_{ac,c}^{CC}$ for every value of $d$. In addition, we\ncan see that for $d < 60$ m and $d<110$ m, $T_{ac,c}^{LT}$ is less than\n$T_{ac,c}^{BC}$ and $T_{ac,c}^{CC}$, respectively. Using (\\ref{total_energy1})\nand Fig. \\ref{fig: Coded_Uncoded}, and noting that the total energy consumption of a certain scheme is proportional\nto the transmission time, it is concluded that the scheme with a lower transmission time\nis not necessarily more energy efficient than that of the scheme with greater $T_{ac}$.\n\n\\begin{figure}[t]\n\\centerline{\\psfig{figure=T_ac_vs_d.eps,width=4.2in}}\n\\caption{Active mode duration of the optimized uncoded and coded MFSK schemes versus transmission distance $d$.} \\label{fig: active}\n\\end{figure}\n\n\n\\textbf{Remark 2:} To make a fair comparison between uncoded and coded schemes,\none would expect to assume a constant active mode duration for all the communication\nschemes. According to $T_{ac,c}=\\frac{T_{ac}}{R_c}$ with $T_{ac}=\\frac{MN}{B \\log_2 M}$,\nthis is achieved by adjusting the modulation order $M$. However, it is\nworth mentioning that the assumption of the same transmission times for all the\nschemes is not a realistic assumption in feasible WSNs where ``autonomous'' sensor\ndevices are powered by limited-lifetime batteries. More precisely, under the\nassumption of the same $T_{ac}$ (or $T_{ac,c}$), the wireless sensor network\nneeds an extra hardware to adjust the constellation size $M$ (in each distance $d$),\nwhich imposes more cost, complexity and power consumption in the network. While,\nour scheme with variable transmission times surpasses the above adaptive\nmodulation constraint using the fact that the optimum constellation size $M$ which\nminimizes the total energy consumption (in each distance $d$) is $\\hat{M}=2$.\n\n\n\n\\section{Conclusion}\\label{conclusion_Ch6}\nIn this paper, we analyzed the energy efficiency of LT coded MFSK in\na proactive WSN over Rayleigh fading channels with path-loss. It was\nshown that the energy efficiency of LT codes is similar to that of\nuncoded MFSK scheme for $d d_T$, LT coded MFSK\noutperforms other uncoded and coded schemes, from the energy\nefficiency point of view. This result follows from the flexibility\nof the LT code to adjust its rate and the corresponding LT coding\ngain to suit instantaneous channel conditions for any transmission\ndistance $d$. This rate flexibility offers strong benefits in using\nLT codes in practical WSNs with dynamic distance and position\nsensors. In such systems and for every value of distance $d$, LT\ncodes can adjust their rates to achieve a certain BER with low\nenergy consumption. The importance of our scheme is that it avoids\nsome of the problems inherent in adaptive coding or Incremental\nRedundancy (IR) systems (channel feedback, large buffers, or\nmultiple decodings), as well as the coding design challenge for\nfixed-rate codes used in WSNs with dynamic position sensor nodes.\nThe simplicity and flexibility advantages of LT codes make the LT\ncode with MFSK modulation can be considered as a \\emph{Green\nModulation\/Coding} (GMC) scheme in dynamic WSNs.\n\nIn this paper, we have shown the significant benefit of rateless codes\n(with focus on the optimized LT codes) for sensor networks over the more\ntraditional fixed-rate codes. Our future research involves selecting the\nbest rate-adaptive code among LT, Raptor, punctured LDPC, and punctured\nTurbo codes; in particular, we are interested to study the performance\nof the Raptor code which has a linear-time encoding versus the non-linear\ncost of the LT code. For this study, a particularly nice feature of the\nLT code is to rapidly optimize the code using a modified EXIT chart\nstrategy introduced in \\cite{Etesami_ITIT0506}.\n\n\n\\appendices\n\n\\section{Asymptotic LT Code Rate}\nThe proof of the remark is straightforward using the notation of\n\\cite{Etesami_ITIT0506} and the bipartite graph concepts in graph\ntheory. Obviously, the output-node degree distribution\n$\\mathcal{O}(x)$ induces a distribution on the input nodes in the\nbipartite graph. Thus, in the asymptotic case of $k \\to \\infty$, we\nhave the input-node degree distribution defined as $\\mathcal{I}(x)\n\\triangleq \\sum_{i=1}^{\\infty}\\nu_i x^i$, where $\\nu_i$ denotes the\nprobability that an input node has a degree $i$. In this case, the\naverage degree of the input and output nodes are computed as\n$\\frac{d \\mathcal{I}(x)}{d x}\\big|_{x=1} \\triangleq\n\\mathcal{I}_{ave}$ and $\\frac{d \\mathcal{O}(x)}{d x}\\big|_{x=1}\n\\triangleq \\mathcal{O}_{ave}$, respectively. Thus, the number of\nedges exiting the input nodes of the bipartite graph, in the\nasymptotic case of $k \\to \\infty$, is $k\\mathcal{I}_{ave}$, which\nmust be equal to $n\\mathcal{O}_{ave}$, the number of edges entering\nthe output nodes in the graph. As a results, the asymptotic LT code\nrate is obtained as $R^{LT}_c= \\frac{k}{n} \\approx\n\\frac{\\mathcal{O}_{ave}}{\\mathcal{I}_{ave}}$, which is a\ndeterministic value for given $\\mathcal{O}(x)$ and $\\mathcal{I}(x)$.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\nIn \\cite{Wi}, Wilson generalized the work of Church-Ellenberg-Farb \\cite{CEF} on FI-modules to the setting of \\(\\FI_\\mathcal{W}\\)-modules, proving strong results about these and finding a wide range of interesting applications. In \\cite{SS2}, Sam-Snowden substantially generalized Wilson's work to that of \\(\\FI_G\\)-modules, for \\(G\\) a virtually polycyclic group, where Wilson's \\(\\FI_{\\text{BC}} = \\FI_{\\mathbb{Z}\/2\\mathbb{Z}}\\). Recall that a group is polycyclic if it has a subnormal series with cyclic factors, and virtually polycyclic if it has a polycyclic subgroup of finite index.\n\nSam-Snowden proved the Noetherian property for \\(\\FI_G\\), and Gan-Li \\cite{GL3} extended this to a proof of actual representation stability in the sense of \\cite{CF}. However, few examples of \\(\\FI_G\\)-modules have been discussed. Here, we extend Wilson's work to exhibit a wide class of examples of \\(\\FI_G\\)-modules. For any virtually polycyclic group \\(G\\), we give applications to the following sequences of representations of \\(W_n = G^n \\rtimes S_n\\) (see \\S3.1 and \\S7 for definitions):\n\\begin{enumerate}\n\\item The homology \\(H_i(\\Conf_n^G(M); \\mathbb{C})\\) of the orbit configuration space \\(\\Conf_n^G(M)\\) associated to the action of \\(G\\) on an open manifold \\(M\\), and the cohomology \\(H^i(\\Conf_n^G(M); \\mathbb{C})\\) in the case where \\(M\\) need not be open but \\(G\\) is finite.\n\\item The rational homotopy groups \\(\\pi_i(\\Conf_n(M))\\) of an open manifold \\(M\\) of dimension \\(\\ge 3\\) with \\(G = \\pi_1(M)\\) virtually polycyclic, or the dual rational homotopy groups \\(\\Hom(\\pi_i(\\Conf_n(M)), \\mathbb{Q})\\) where \\(M\\) need not be open but \\(G\\) is finite.\n\\item The cohomology \\(H^i(\\FR(G^{\\ast n}); \\mathbb{C})\\) of the Fouxe-Rabinovitch group of \\(G\\), where each \\(H^i(G; \\mathbb{Q})\\) is finite-dimensional.\n\\end{enumerate}\nFor the specific case \\(G = \\mathbb{Z}\/d\\mathbb{Z}\\), the resulting automorphism groups \\(W_n = \\left(\\mathbb{Z}\/d\\mathbb{Z}\\right)^n \\wr S_n\\) are the so-called main series of complex reflection groups. These directly generalize the Weyl groups of type \\(BC_n\\) of Wilson's paper, for which \\(d = 2\\). We give applications to the following sequence of representations of \\(W_n\\) (see \\S4.1, \\S4.2, \\S4.3 for definitions):\n\\begin{enumerate}\n\\setcounter{enumi}{3}\n\\item The cohomology \\(H^i(P(d,n); \\mathbb{C})\\) of the pure monomial braid group.\n\\item The graded pieces \\(\\gr P(d,n)^i\\) of the associated graded Lie algebra of the pure monomial braid group.\n\\item The graded pieces \\(\\mathcal{C}_J^{(r)}(n)\\) of the complex-reflection diagonal coinvariant algebra.\n\\end{enumerate}\nOur work implies the following results about these sequences. Recall that the irreducible representations of \\(W_n\\) are parameterized by the partition-valued functions \\(\\underline \\lambda\\) on the irreducible representations of \\(G\\) with \\(\\|\\underline \\lambda\\| = n\\). Denote the irreducible representation of \\(W_n\\) associated to \\(\\underline \\lambda\\) by \\(L(\\underline \\lambda)\\). Given partitions \\(\\lambda_1, \\dots, \\lambda_k\\) with \\(|\\lambda_1| + \\dots + |\\lambda_k| = n\\), and irreducible representations \\(\\chi_1, \\dots, \\chi_k\\) of \\(G\\), we will write \\(\\underline\\lambda = ( {\\lambda_1}_{\\chi_1}, \\dots, {\\lambda_2}_{\\chi_2})\\) for the partition-valued function \\(\\underline\\lambda\\) with \\(\\underline\\lambda(\\chi_i) = \\lambda_i\\).\n\nLet \\(c(G)\\) be the set of conjugacy classes of \\(G\\). Define a \\emph{character polynomial} to be a polynomial in \\(c(G)\\)-labeled cycle-counting functions. This is a class function on \\(W_n\\). (We will define this and other terminology more precisely in \\S2.2).\n\\begin{thm}[\\bfseries Polynomiality of characters and representation stability]\nSuppose \\(G\\) is finite, and let \\(\\{V_n\\}\\) be any of the sequences 1-6 above. Then:\\begin{enumerate}\n\\item The characters of \\(V_n\\) are given by a single character polynomial for all \\(n \\gg 0\\).\n\\item The multiplicity of each irreducible \\(W_n\\)-representation in \\(V_n\\) is eventually independent of \\(n\\), and \\(\\dim V_n\\) is eventually polynomial in \\(n\\). \n\nMoreover, for sequences 3 and 4, we can explicity bound when this stabilization occurs.\n\\end{enumerate}\n\\end{thm}\nFor example, for sequence 4, once \\(n \\ge 2\\) we obtain:\n\\[H^1(P(d,n); \\mathbb{C}) = L((n)_{\\chi_0}) \\oplus L((n-1,1)_{\\chi_0}) \\oplus \\bigoplus_{\\chi \\in \\Irr(G)} L((n-2)_{\\chi_0}, (2)_\\chi) \\]\nIn particular, our results on \\(H^i(\\FR(\\Gamma^{\\ast n}); \\mathbb{C})\\) answer a question of Wilson, posed in \\cite[\\S 7.1.1]{Wi}, of how to generalize her Thm 7.3 to a general Fouxe-Rabinovitch group. Our results on \\(\\pi_i(\\Conf_n(M))\\) both confirm and correct a sketch given by Kupers-Miller \\cite{KM} about the homotopy groups of configuration spaces of non-simply-connected manifolds, and answer their question about obtaining a form of representation stability for finite \\(G\\).\n\\subsection*{Infinite \\(G\\): \\(K_0\\)-stability and finite presentation degree}\nWhen \\(G\\) is infinite virtually polycyclic, we still obtain a form of representation stability. For a ring \\(R\\), recall that the Grothendieck group \\(G_0(R)\\) is the free abelian group on finitely-generated \\(R\\)-modules, modulo the relations that the center of a short exact sequence is equal to the sum of the left and right terms. \\(K_0(R)\\) is defined similarly, but only for projective \\(R\\)-modules; these two groups coincide when \\(R = k[G]\\) for \\(G\\) virtually polycyclic. Say that a sequence \\(V_n\\) of \\(W_n\\)-representations satisfies \\emph{\\(K_0\\)-stability} if the decomposition of \\([V_n]\\) in the \\emph{Grothendieck group} \\(K_0(k[W_n])\\) is eventually independent of \\(n\\). We apply this notion to 1-3 above, and to the following two more examples:\n\\begin{enumerate} \\setcounter{enumi}{6}\n\\item The homology \\(H_i(P_{\\widetilde{A}_{n-1}}; \\mathbb{Q})\\) of the affine pure braid group of type \\(\\widetilde{A}_{n-1}\\), for which \\(G = \\mathbb{Z}\\).\n\\item The homology \\(H_i(P_{\\widetilde{C}_{n}}; \\mathbb{Q})\\) of the affine pure braid group of type \\(\\widetilde{C}_{n}\\), for which \\(G = \\mathbb{Z} \\rtimes \\mathbb{Z}\/2\\).\n\\end{enumerate}\n\\begin{thm}[\\bfseries \\(K_0\\)-stability]\nLet \\(\\{V_n\\}\\) be any of the sequences above (1-3,7,8) for which \\(G\\) is infinite virtually polycyclic. Then \\(V_n\\) satisfies \\(K_0\\)-stability.\n\\end{thm}\nFor example, if \\(U = k[\\mathbb{Z}] = \\Ind_{\\mathbb{Z}\/2}^{\\mathbb{Z} \\rtimes \\mathbb{Z}\/2} k\\), then in the notation of \\S2.3,\n\\[ [H_1(P_{\\widetilde{C}_{n}}; \\mathbb{Q})] = [L((n)_U)],\\;\\; \\forall n \\ge 1. \\]\nNote that in general we certainly lose information in passing to \\(K_0\\). For instance, in the above example, we actually explicitly determine in \\S6.(\\ref{affC}) that \\(H_1(P_{\\widetilde{C}_{n}}; \\mathbb{Q})\\) is the direct sum of three induced representations, two of which are killed in passing to \\(K_0\\), and the third giving the term \\([L((n)_U)]\\) above.\n\nThis is the first representation stability result obtained for infinite discrete groups (recall that \\cite{CF} obtained representation stability results for Lie groups, whose representation theory is much more tractable). \n\nFurthermore, for a totally general group \\(G\\), we still obtain a weak form of finite generation. Say that an \\(\\FI_G\\)-module is \\emph{presented in finite degree} if it is finitely-generated and if the relations for the generators also live in finite degree. Ramos \\cite{Ra2} and Li \\cite{Li} have proven that the category of \\(\\FI_G\\)-modules that are presented in finite degree is abelian. This allows us to carry over arguments about spectral sequences to conclude that cohomology is generated in finite degree. In particular, we have:\n\\begin{thm}[\\bfseries Finite presentation degree]\nThe cohomology \\(H^i(\\Conf^G_n(M); \\mathbb{Q})\\) and the dual rational homotopy groups \\(\\Hom(\\pi_i( \\Conf_n(M)); \\mathbb{Q})\\) are \\(\\FI_G\\)-modules presented in finite degree.\n\\end{thm}\n\n\\subsection*{Application: arithmetic statistics for Gauss sums}\nIn the sequel \\cite{Ca2}, we apply the theory of representation stability for \\(\\FI_G\\)-modules to obtain results about weighted point-counts for polynomials over finite fields. For example, we are able to obtain results such as the following. Let \\(\\chi\\) be a character of \\(\\mathbb{Z}\/(q-1)\\). Define the character polynomial \\(X_i^\\chi = \\sum_{g \\in G} \\chi(g) X^g_i\\). Then\n\\begin{gather} \\begin{aligned} \\label{X2_pcount}\n\\lim_{n \\to \\infty} &\\sum_{f \\in \\UConf_n(\\mathbb{F}_q^*)} \\sum_{\\deg(p) = 2} \\chi(\\text{root}(p)) \\\\\n&= \\sum_i (-1)^i \\frac{\\langle X_2^{\\overline \\chi}, H^i(\\Conf^{\\mathbb{Z}\/(q-1)}(\\mathbb{C}^*); \\mathbb{Q}(\\zeta_{q-1}))\\rangle}{q^i} = \\frac{1}{2q} + \\frac{2}{q^2} + \\cdots\n\\end{aligned} \\end{gather}\nThat is, the average value of the Gauss sum obtained by applying \\(\\chi\\) to the \\emph{quadratic} factors of \\(f\\), across all \\(f \\in \\UConf_n(\\mathbb{F}^*_q)\\), is equal to the series on the right obtained by looking at the inner product of the character polynomial \\(X_2^{\\overline \\chi}\\) with \\(H^i(\\Conf^{\\mathbb{Z}\/(q-1)}(\\mathbb{C}^*); \\mathbb{Q}(\\zeta_{q-1}))\\). The fact that the sum on the right is independent of \\(n\\) for large \\(n\\) is precisely a consequence of representation stability for the \\(\\FI_G\\)-module \\(H^i(\\Conf^{\\mathbb{Z}\/(q-1)}(\\mathbb{C}^*))\\).\n\n\\subsection*{Acknowledgements}\nI would like to thank Nir Gadish, Weiyan Chen, Jesse Wolfson, John Wiltshire-Gordon, Eric Ramos, and Jeremy Miller for helpful conversations. Also, I am very grateful to my advisor, Benson Farb, for all his guidance throughout the process of working on, writing, and revising my first paper.\n\n\\section{\\(\\FI_G\\)-modules and their properties}\nIf \\(G\\) is a group, and \\(R\\) and \\(S\\) sets, define a \\emph{\\(G\\)-map} \\((a, (g_i)): R \\to S\\) to be a pair \\(a: R \\to S\\) and \\((g_i) \\in G^R\\). If \\((b, (h_j)): S \\to T\\) is another \\(G\\)-map, their composition is \\((b \\circ a, (g_i \\cdot h_{a(i)}))\\). Let \\(\\FI_G\\) be the category with objects finite sets and morphisms \\(G\\)-maps with the function \\(a\\) injective. This is clearly equivalent to the full subcategory with objects the sets \\([n] = \\{1, \\dots, n\\}\\). Note that the automorphism group of \\([n]\\) is\n\\[W_n := G \\wr S_n = G^n \\rtimes S_n\\]\nAn \\emph{\\(\\FI_G\\)-module over \\(k\\)} is just a functor \\(V: \\FI_G \\to k\\Mod\\); when \\(k\\) is clear, we simply write \\(\\FI_G\\)-modules. These form a category, called \\(\\FI_G\\Mod\\). Thus an \\(\\FI_G\\)-module \\(V\\) is a sequence of \\(W_n\\)-representations \\(V_n\\), with maps \\(V_n \\to V_{n+1}\\) satisfying certain coherency conditions. Say that \\(V\\) is \\emph{finitely generated} if there is a finite set of elements \\(v_1, \\dots, v_n \\in V\\) such that the smallest \\(\\FI_G\\)-submodule containing the \\(v_i\\) is all of \\(V\\).\n\nFor \\(m \\ge 0\\), define the ``free'' \\(\\FI_G\\)-module \\(M(m)\\) by setting\n\\[M(m)_n = \\begin{cases} 0 & n < m \\\\ k[\\Hom_{\\FI_G}([m], [n])] & n \\ge m \\end{cases}\\]\nRecall that for an \\(\\FI_G\\)-module \\(V\\) and an element \\(v \\in V_m\\), we can also characterize the submodule generated by \\(v\\) as the image of the map\n\\[ M(m) \\to V, \\;\\; f \\in \\Hom_{\\FI_G}([m], [n]) \\mapsto f_* v \\]\nWe can thus characterize finitely generated \\(\\FI_G\\)-modules as those \\(V\\) that admit a surjection \\(\\bigoplus_{i=1}^N M(m_i) \\twoheadrightarrow V\\).\n\n\\subsection{Noetherianity and representation stability}\n\nRecall that a group is \\emph{polycyclic} if it has a composition series with cyclic factors, and is \\emph{virtually polycyclic} if it has a polycyclic subgroup of finite index. Virtually polycyclic groups are of interest, among other things, because they are the only known groups to have Noetherian group rings, and are conjectured to be the only such groups. In \\cite[Cor 1.2.2]{SS2}, Sam-Snowden proved that if \\(G\\) is a virtually polycyclic group, then \\(\\FI_G\\Mod\\) is Noetherian over any Noetherian ring \\(k\\). That is, they proved that any finitely generated \\(\\FI_G\\)-module has all its submodules finitely generated. The crucial property used was that the group ring \\(k[G]\\) is Noetherian.\n\nFor FI-modules, the most important consequence of finite generation is representation stability, and for \\(G\\) finite, Gan-Li proved in \\cite[Thm 1.10]{GL3} that this holds for \\(\\FI_G\\) as well. Technically there are three parts to representation stability according to the definition given in \\cite{CF}, but the first two parts (``surjectivity'' and ``injectivity'') follow straightforwardly from the definition of being finitely generated. It is the third part, ``multiplicity stability'', that is really the most interesting, and which we will now describe.\n\nTake \\(G\\) finite and \\(k\\) a \\emph{splitting field} of characteristic 0 for \\(G\\), that is, a field over which all its irreducible representations over \\(\\mathbb{C}\\) are defined. Let \\(\\Irr(G)\\) denote the set of isomorphism classes of irreducible representations of \\(G\\). Let \\(\\underline \\lambda\\) be a partition-valued function on \\(\\Irr(G)\\). Put \\(|\\underline\\lambda| = (|\\underline\\lambda(\\chi_1)|, \\dots, |\\underline\\lambda(\\chi_r)|)\\), and \\(\\|\\underline\\lambda\\|\\) for the norm of this partition. Then if \\(\\|\\underline\\lambda\\| = n\\), there is an associated irreducible representation of \\(W_n\\):\n\\[ L(\\underline \\lambda) = \\Ind^{W_n}_{W_{|\\underline\\lambda|}} \\left(M_1^{\\otimes \\underline\\lambda(\\chi_1)} \\otimes E(\\underline\\lambda(\\chi_1))\\right) \\otimes \\cdots \\otimes \\left(M_r^{\\otimes \\underline\\lambda(\\chi_r)} \\otimes E(\\underline\\lambda(\\chi_1))\\right) \\]\nwhere \\(W_\\mu = W_{\\mu(1)} \\times \\cdots W_{\\mu(l)}\\),\nand that these comprise all the irreducible representations of \\(W_n\\) up to isomorphism. Extend \\(\\underline{\\lambda}\\) to \\(n \\ge m + \\underline{\\lambda}(\\chi_0)_1\\) as follows:\n\\[\n\\underline{\\lambda}[n](\\chi) = \\begin{cases} \\left(n - |\\underline{\\lambda}|, \\underline{\\lambda}(\\chi_0)\\right) & \\text{if } \\chi = \\chi_0 \\\\ \\underline{\\lambda}(\\chi) & \\text{otherwise} \\end{cases}\n\\]\nWriting \\(L(\\underline{\\lambda})_n\\) for the irreducible representation corresponding to \\(\\underline{\\lambda}[n]\\), multiplicity stability for an \\(\\FI_G\\)-module \\(V\\) says that the decomposition into irreducibles has multiplicities independent of \\(n\\) for large \\(n\\):\n\\[\nV_n = \\bigoplus_{\\underline{\\lambda}} L(\\underline{\\lambda})_n^{\\oplus c(\\underline{\\lambda})} \\,\\text{ for all } n \\ge N\n\\]\nwhere we call \\(N\\) the \\emph{stability degree} of \\(V\\). In particular, when \\(G\\) is trivial, we recover (uniform) multiplicity stability for FI-modules in the sense of \\cite[Defn 2.6]{CF}, and when \\(G = \\mathbb{Z}\/2\\mathbb{Z}\\), we recover (uniform) multiplicity stability for \\(\\FI_{\\text{BC}}\\)-modules in the sense of \\cite[Defn 2.6]{Wi}.\n\n\\subsection{Projective resolutions and character polynomials}\nFor \\(G\\) finite, \\cite{SS2} and \\cite{GL3} actually obtain a deeper structural result that implies representation stability, and this result has other important consequences for us. To state it, define a \\emph{torsion} \\(\\FI_G\\)-module to be one with \\(V_n \\ne 0\\) for only finitely many \\(n\\), and say that an \\(\\FI_G\\)-module is projective if it is a projective object in the category \\(\\FI_G\\Mod\\). Gan-Li's result can then be stated as follows.\n\n\\begin{prop}[{\\cite[Thm 1.6]{GL3}}]\nFor any finitely generated \\(\\FI_G\\)-module \\(V\\), with \\(G\\) finite, there is a finite resolution of \\(\\FI_G\\) modules\n\\[0 \\to V \\to T^1 \\oplus P^1 \\to T^2 \\oplus P^2 \\to \\dots \\to T^n \\oplus P^k \\to 0\\]\nwith each \\(T^i\\) torsion and each \\(P^i\\) projective.\n\\end{prop}\n\nIn particular, this resolution's existence means that for \\(n \\gg 0\\) there is a resolution of \\(W_n\\)-representations\n\\begin{equation} \\label{fin_res}\n0 \\to V_n \\to P^1_n \\to \\dots \\to P^k_n \\to 0\n\\end{equation}\n\nThis are powerful because, as in \\cite{CEF}, we have strong control over the structure of projective \\(\\FI_G\\)-modules. Namely, let \\(\\Res: \\FI_G \\to W_i\\) be the restriction to a single group. This functor has a left adjoint \\(\\Ind^{\\FI_G}: W_i \\to \\FI_G\\) given by\n\\[\\Ind^{\\FI_G}(V)_{i+j} = \\Ind_{W_i \\times W_j}^{W_{i+j}} V \\boxtimes k\\]\nThen \\(\\Ind^{\\FI_G}(V)\\) is a projective \\(\\FI_G\\)-module whenever \\(V\\) is a projective \\(W_i\\)-module, and tom Dieck \\cite[Prop 11.18]{tomD} proved that any projective \\(\\FI_G\\)-module is of this form (in fact, for a large class of category representations). Following Ramos \\cite{Ra1}, define a \\emph{relatively projective} module to be a direct sum of any such induced modules, even if the \\(W_i\\) are not projective \\(W_n\\)-representations. Finitely-generated (relatively) projective \\(\\FI_G\\)-modules thus have a compact description, as direct sums of induced modules from a finite list of \\(W_n\\)-representations, even for infinite \\(G\\).\n\nIn fact, Ramos \\cite{Ra1} has found an effective bound for when \\(n\\) occurs in (\\ref{fin_res}). To describe it, we need to refine our notion of finite generation. As we said, finitely generated modules are those that admit a surjection \\(\\bigoplus_{i=1}^N M(d_i) \\twoheadrightarrow V\\). Say that such a \\(V\\) is \\emph{generated in degree} \\(\\le d = \\max_i \\{d_i\\}\\). Next, suppose there is an exact sequence\n\\[ 0 \\to K \\to M \\to V \\to 0\\]\nwith \\(M\\) relatively projective and \\(K\\) generated in degree \\(\\le r\\). Then say that \\(V\\) is has \\emph{related in degree} \\(\\le r\\).\n\nRamos \\cite[Thm C]{Ra1} then says that if \\(V\\) is a finitely-generated \\(\\FI_G\\)-module generated in degree \\(d\\) and related in degree \\(r\\), then the above resolution (\\ref{fin_res}) holds whenever \\(n \\ge r + \\min\\{r, d\\}\\). Notice that if \\(V\\) is already relatively projective, then \\(r = 0\\) and this is sharp.\n\nIn particular, when \\(G\\) is finite, the resolution (\\ref{fin_res}) implies representation stability: as Gan-Li verify in \\cite[Thm 1.10]{GL3}, the individual projective modules \\(\\Ind^{\\FI_G}(V_m)\\) satisfy representation stability, with stability degree \\(\\le 2m\\). By semisimplicity of each \\(k[G_n]\\), \\(V\\) therefore satisfies representation stability as well. Furthermore, this resolution provides a quick proof of the existence of character polynomials for \\(\\FI_G\\), as follows.\n\nRecall that for a \\emph{character polynomial} for \\(S_n\\) is an element of \\(\\mathbb{Q}[X_1, X_2, \\dots]\\), which we think of as a class function on \\(S_n\\), where \\(X_i\\) counts the number of \\(i\\)-cycles of a permutation. \\cite[Thm 3.3.4]{CEF} prove that any finitely generated \\(\\FI\\)-module is eventually given by a single character polynomial. We generalize this and Wilson's \\cite[Thm 5.15]{Wi} result for \\(G = \\mathbb{Z}\/2\\) as follows.\n\\begin{thm}[\\bfseries Character polynomials for \\(\\FI_G\\)]\n\\thlabel{charpoly}\nLet \\(G\\) be a finite group and \\(k\\) a splitting field for \\(G\\) of characteristic 0. If \\(V\\) is a finitely generated \\(\\FI_G\\)-module over \\(k\\), generated in degree \\(m\\) and related in degree \\(r\\), then there is a polynomial\n\\[P_V \\in k\\left[\\{X_i^C \\mid i \\ge 1, C \\text{ is a conjugacy class of } G\\}\\right]\\]\nof degree \\(\\le m\\), where \\(X_i^C\\) is the class function counting the number of \\(C\\)-labeled \\(i\\)-cycles in the conjugacy class of \\(g\\), so that for all \\(n \\ge r + \\min(m,r)\\)\n\\[\\chi_{V_n}(g) = P_V(g).\\]\n\\end{thm}\n\\begin{proof}\nIf \\(V\\) is a representation of \\(W_m\\), we can explicitly compute the character of the projective \\(\\FI_G\\)-module \\(\\Ind^{\\FI_G}(V)\\). This calculation is done in \\cite[Lem 5.14]{Wi} for \\(G = \\mathbb{Z}\/2\\), and her proof applies essentially verbatim, but enough small details are different that it is easier to just give the adapted proof than to describe the necessary changes.\n\nThe character of the induced representation \\(\\Ind^{\\FI_G}(V)_n = \\Ind_{W_m \\times W_{n-m}}^{W_n} V \\boxtimes k\\) is\n\\begin{align*}\n\\chi_{\\Ind^{\\FI_G}(V)_n}(w) &= \\sum_{\\substack{\\{\\text{cosets } C\\, \\mid\\, w \\cdot C = C\\} \\\\ \\text{any } s \\in C}} \\chi_{V \\boxtimes k}(s^{-1} w s) \\\\\n&= \\sum_{\\substack{\\{\\text{cosets } C\\, \\mid\\, w \\cdot C = C\\} \\\\ \\text{any } s \\in C}} \\chi_V(p_m(s^{-1} w s))\n\\end{align*}\nwhere \\(p_m: W_m \\times W_{n-m} \\to W_m\\) is the projection, and where the sum is over all cosets in \\(W_n \/ (W_m \\times W_{n-m})\\) that are stabilized by \\(w\\), equivalently, those cosets \\(C\\) such that \\(s^{-1}ws \\in W_m \\times W_{n-m}\\) for any \\(s \\in C\\).\n\nAn element \\(w \\in W_n\\) can be conjugated in \\(W_m \\times W_{n-m}\\) precisely when its \\(c(G)\\)-labeled cycles can be split into a set of cycles of total length \\(m\\), and a set of cycles of total length \\((n-m)\\). If we fixed a labeled partition \\(\\underline \\lambda\\) of \\(m\\), then the cycles of \\(w\\) can be factored into an element \\(w_m\\) of labeled cycle type \\(\\underline \\lambda\\) and its complement \\(w_{n-m}\\) in the following number of ways (possibly 0):\n\\[\n\\prod_{C \\in c(G)} \\binom{X^C_1}{n_1(\\underline{\\lambda}(C))} \\binom{X^C_2}{n_2(\\underline{\\lambda}(C))} \\cdots \\binom{X^C_m}{n_m(\\underline{\\lambda}(C))}\n\\]\nwhere \\(n_r(\\mu)\\) is the number of \\(r\\)'s in \\(\\mu\\). Each such factorization of \\(w\\) corresponds to a coset \\(C \\in W_n \/ (W_m \\times W_{n-m})\\) that is stabilized by \\(w\\). For any representative \\(s \\in C\\), \\(p_m(s^{-1} w s)\\) has labeled cycle type \\(\\underline \\lambda\\). So we conclude that\n\\[\n\\chi_{\\Ind^{\\FI_G}(V)_n} = \\sum_{|\\underline{\\lambda}| = m} \\chi_V(\\underline{\\lambda}) \\prod_C \\binom{X^C_1}{n_1(\\underline{\\lambda}(C))} \\binom{X^C_2}{n_2(\\underline{\\lambda}(C))} \\cdots \\binom{X^C_m}{n_m(\\underline{\\lambda}(C))}\n\\]\nwhere the left-hand-side is manifestly a fixed character polynomial independent of \\(n\\). Notice that this character polynomial has degree \\(m\\). The general result therefore again follows from (\\ref{fin_res}) and Ramos \\cite[Thm C]{Ra1} by semisimplicity.\n\\end{proof}\n\nIn particular, by taking \\(g = e\\) in \\thref{charpoly}, we see that \\(\\dim V_n\\) is given by a single polynomial for \\(n \\ge r + \\min(m,r)\\), which is \\cite[Thm D]{Ra1}.\n\n\\subsection{Virtually polycyclic \\(G\\) and \\(K_0\\)-stability}\nThe above results only apply when \\(G\\) is finite, so that the group ring \\(k[G]\\) is semisimple. We would like to have a good analogue of representation stability for infinite virtually polycyclic \\(G\\). The answer is to pass to the Grothendieck group, where we essentially impose semisimplicity by fiat.\n\nRecall that, for a ring \\(R\\), the Grothendieck group \\(G_0(R)\\) is defined to be the free abelian group generated by isomorphism classes \\([M]\\) of finitely-generated \\(R\\)-modules \\(M\\), modulo the relation that for any short exact sequence\n\\[ 0 \\to M \\to N \\to M' \\to 0,\\;\\text{ we have }\\;[N] = [M] + [M']\\]\nThe Grothendieck group \\(K_0(R)\\) is defined similarly, except that we only take finitely-generated \\emph{projective} \\(R\\)-modules. The map \\(K_0(R) \\to G_0(R)\\), \\([P] \\to [P]\\) is well-defined and is called the \\emph{Cartan map}.\n\nHere we will be taking \\(R = k[W_n]\\), the group ring of \\(W_n = G \\wr S_n\\) with \\(G\\) a virtually polycyclic group. Since \\(W_n\\) is itself virtually polcyclic, it is then known that the Cartan map is an isomorphism, so the two notions of Grothendieck group coincide. We will use the notation \\(K_0(W_n) := K_0(k[W_n]) \\cong G_0(k[W_n])\\), as much to avoid clashing with the group \\(G\\), but we abuse notation by writing \\([M] \\in K_0(W_n)\\) for any finitely-generated \\(W_n\\)-module \\(M\\), which makes sense since the Cartan map is an isomorphism.\n\nGrothendieck groups of virtually polycyclic groups have been much studied and are well understood; for a good overview, see \\cite[Ch. 8]{Pa}. For one thing, such groups are always torsion-free and finitely generated, and thus isomorphic to \\(\\mathbb{Z}^r\\). One powerful result is \\emph{Moody's induction theorem} \\cite{Mo}, which says that for \\(G\\) virtually polycyclic, \\(K_0(k[G])\\) is generated by induced modules of the finite subgroups of \\(G\\). This means that, at the level of \\(K_0\\), the representation theory of \\(G \\wr S_n\\) for \\(G\\) virtually polycyclic works in the same way as for \\(G\\) finite that was described in \\S2.1, as follows. \n\nThe monoid \\(K^+_0(G)\\) of \\emph{positive} classes---that is, those actually represented by honest, and not just virtual, representations---is isomorphic to \\(\\mathbb{N}^r\\), and thus has a unique basis \\(X = \\{\\chi_1, \\dots, \\chi_r\\}\\). Pick representative \\(G\\)-modules \\(M_1, \\dots, M_r\\), so that \\([M_i] = \\chi_i\\). If \\(\\underline \\lambda\\) is a partition-valued function on \\(X\\), let \\(L(\\underline \\lambda)\\) be the associated representation of \\(W_n\\) defined in \\S2.1, using the \\(M_i\\) as the representations associated to \\(\\chi_i\\). Of course, this is only defined up to a choice of \\(\\{M_i\\}\\), but it does give a well-defined element \\([L(\\underline\\lambda)] \\in K_0(G)\\). Moody's induction theorem then has the following consequence:\n\n\\begin{prop}\nThe collection \\(\\{[L(\\underline \\lambda)] : \\|\\underline\\lambda\\| = n\\}\\) forms a basis for \\(K_0(W_n)\\).\n\\end{prop}\n\\begin{proof}\nThe fact that the \\([L(\\underline\\lambda)\\)]'s span \\(K_0(W_n)\\) is the actual consequence of Moody's induction theorem. For linear independence---that is, injectivity of the map \\(\\mathbb{Z}\\langle [L(\\underline \\lambda)] \\rangle_{\\underline \\lambda} \\to K_0(W_n)\\)---see e.g. \\cite[Exer 35.3]{Pa}.\n\\end{proof}\n\nThe reason \\(K_0\\) is relevant to us is because of the following generalization of the resolution (\\ref{fin_res}) to virtually polycyclic \\(G\\), due to Nagpal-Snowden:\n\\begin{thm}[Nagpal-Snowden \\cite{NS}]\nIf \\(G\\) is virtually polycyclic, and \\(V\\) is a finitely-generated \\(\\FI_G\\)-module, then for \\(n\\) sufficiently large, there is a filtration\n\\[0 = V^{(0)}_n \\subset V^{(1)}_n \\subset \\cdots \\subset V^{(m)}_n = V_n\\]\nsuch that the successive quotients \\(V_n^{(k+1)}\/V_n^{(k)}\\) are of the form \\(\\bigoplus_i \\Ind^{\\FI_G}(W_i)_n\\).\n\\end{thm}\nWe are now ready to formulate the relevant generalization of representation stability. For \\(G\\) a virtually polycyclic group, say that an \\(\\FI_G\\)-module \\(V\\) satisfies \\emph{\\(K_0\\)-stability} if for all sufficiently large \\(n\\), there is a decomposition\n\\[ [V_n] = \\sum_{\\underline\\lambda} c(\\underline\\lambda) [L(\\underline\\lambda[n])]\\]\nwhere \\(c(\\underline\\lambda)\\) does not depend on \\(n\\). We then obtain the following.\n\n\\begin{thm}[\\bfseries \\(K_0\\)-stability]\nIf \\(G\\) is virtually polycyclic and \\(V\\) is a finitely-generated \\(\\FI_G\\)-module, then \\(V\\) satisfies \\(K_0\\)-stability.\n\\end{thm}\n\\begin{proof}\nBy Theorem 2.4, for \\(n\\) sufficiently large, we can write\n\\[ [V_n] = \\sum_i [\\Ind^{\\FI_G}(V_i)_n] - \\sum_j [\\Ind^{\\FI_G}(W_j)_n] \\]\nfor some collection of representations \\(\\{V_i\\}\\), \\(\\{W_j\\}\\), independent of \\(n\\). It therefore suffices to verify that the individual terms \\(\\Ind^{\\FI_G}L(\\underline\\lambda)\\) satisfy \\(K_0\\)-stability.\n\nBut now the proof given in \\cite[Thm 1.10]{GL3} of representation stability of \\(\\Ind^{\\FI_G} L(\\underline\\lambda)\\) applies verbatim, since it does not use anything about the group \\(G\\); it just uses Pieri's formula and other facts about the representation theory of \\(S_n\\).\n\\end{proof}\n\n\\subsection{Arbitrary \\(G\\) and finite presentation degree}\nUntil recently, the Noetherian property was seen as the lynchpin of the theory of \\(\\FI\\)-modules and related categories. However, coming out of the work of Church-Ellenberg \\cite{CE} on homological properties of \\(\\FI\\)-modules, a new perspective has emerged that shifts the emphasis from finite generation of modules to finite \\emph{presentation degree} of modules, e.g. in the work of Ramos \\cite{Ra2} and Li \\cite{Li}. In one sense, this perspective is more ``constructive'', because possessing the knowledge of both the degree of generation \\emph{and} the degree of relation of a module gives us quantitative control over various stability properties of the module, as we have seen. The Noetherian property then tells us that \\emph{any} finitely generated module is necessarily finitely presented, which is an important fact but no longer at the absolute center of the theory. Appealing to Noetherianity is also necessarily ineffective, since we are no longer able to say what the relation degree is, and thus lose effective bounds on stability.\n\nAt the same time, this shift in perspective allows us to expand our scope to situations where there is no hope of finite generation. For example, we will see later on examples of \\(\\FI_G\\)-modules \\(V\\) that are not finitely generated simply because the individual pieces \\(V_n\\) are not finitely-generated \\(W_n\\)-representations. Nevertheless, we will be able to prove that \\(V\\) is still generated in finite degree, in the sense that all the generators of \\(V\\) (an infinite number) live only in \\(V_1, \\dots V_m\\) for some \\(m\\), \\emph{and} that \\(V\\) is related in finite degree. Of course, in a sense such \\(V\\) are much \\emph{less} constructive then anything in the finitely-generated world.\n\nThis shift also allows us to leave behind the requirement that \\(G\\) be virtually polycyclic. Indeed, as we have seen, this requirement is based in the fact that, for any kind of Noetherianity to get off the ground, we certainly need the ring \\(k[G]\\) to be Noetherian, which as far as we know is only true when \\(G\\) is virtually polycyclic. However, once we have accepted that modules can be infinitely generated, and only care about the \\emph{degree} that that they are generated (and related) in, we can leave this need behind.\n\nThe central result that informs this perspective is the following, proved simultaneously by Ramos and Li:\n\\begin{thm}[{\\cite[Thm B]{Ra2}, \\cite[Prop 3.4]{Li}}] \\thlabel{coh}\nFor any group \\(G\\), the category of \\(\\FI_G\\)-modules presented in finite degree is abelian.\n\\end{thm}\n\nThis is the analogue of Noetherianity for finite presentation degree, since it allows us to argue that kernels of maps between \\(\\FI_G\\)-modules presented in finite degree are still presented in finite degree, and therefore for example to chase being presented in finite degree through a spectral sequence. Indeed, from a certain point of view \\thref{coh} is the fundamental fact, and Noetherianity as we have seen it is so far is just a consequence of \\thref{coh} and the fact that the individual group rings \\(k[G]\\) are Noetherian.\n\nWhat we lose at this level of generality is any ability to refer back to stability results in terms of things like ``representation stability'' or ``stability of character polynomials'' that are not couched explicitly in terms of \\(\\FI_G\\)-modules. All we can say is that the \\(\\FI_G\\)-modules in question are presented in finite degree, which perhaps is less interesting to someone who only cares about the individual \\(W_n\\)-representations \\(V_n\\). At the same time, since these are infinitely-generated representations of infinite groups, it is hard to say much about the individual representations.\n\n\\subsection{\\(\\FI_G\\sharp\\)-modules}\nThe classification of projective modules provided above means that even when \\(G\\) is infinite, if an \\(\\FI_G\\) module \\(V\\) is projective, then there is still a compact description of the representation theory of \\(V\\). We would therefore like to be able to determine when an \\(\\FI_G\\)-module is projective, so that it has such a description. \\cite{CEF} provide just such a method: they define a category \\(\\FI\\sharp\\) with an embedding \\(\\FI \\hookrightarrow \\FI\\sharp\\) so that \\(\\FI\\sharp\\)-Mod is precisely the category of projective \\(\\FI\\)-modules, so a module is projective just when it extends to an \\(\\FI\\sharp\\)-module. Their construction and proof of equivalence carry over to the setting of \\(\\FI_G\\), as Wilson \\cite{Wi} proved for the case \\(G = \\mathbb{Z}\/2\\).\n\nSo define \\(\\FI_G\\sharp\\) to be the category of \\emph{partial morphisms} of \\(\\FI_G\\): the objects are still finite sets, but a map \\(X \\to Y\\) is given by a pair \\((Z, f)\\), where \\(Z \\subset X\\) and \\(f: Z \\to Y\\) is a \\(G\\)-map. Composition of morphisms is defined by pullback, i.e. with the domain the largest set on which the composition is defined. Then there is a natural structure of \\(\\Ind^{\\FI_G}(V)\\) as an \\(\\FI_G\\sharp\\)-module. Furthermore, we obtain the following.\n\\begin{prop} \\thlabel{sharp_decomp}\nAny \\(\\FI_G\\sharp\\) module is isomorphic to \\(\\,\\bigoplus_i \\Ind^{\\FI_G}(W_i)\\) for some representations \\(W_i\\) of \\(G_i\\).\n\\end{prop}\n\\begin{proof}\nThis was proved for \\(G\\) trivial in \\cite[Thm 4.1.5]{CEF}, and for \\(G = \\mathbb{Z}\/2\\) in \\cite[Thm 4.42]{Wi}. As Wilson explains, the proof in \\cite{CEF} applies almost verbatim: the only change that needs to be made is to the definition of the endomorphism \\(E: V \\to V\\), which should be defined as follows, for \\(m \\ge n\\),\n\\begin{align*}\nE_m&: V_m \\to V_m \\\\\nE_m& = \\sum_{\\substack{S \\subset [m] \\\\ |S| = n}} I_S, \\;\\; \\text{where } I_S = (S, \\iota) \\in \\Hom_{\\FI_G\\sharp}([m], [m]) \\text{ with } \\iota: S \\hookrightarrow [m] \\text{ the inclusion}.\n\\end{align*}\n\\end{proof}\n\n\\begin{cor}\n\\thlabel{sharp_range}\nIf \\(V\\) is an \\(\\FI_G\\sharp\\)-module generated in degree \\(m\\), then \\(\\chi_V\\) is given by a single character polynomial of degree \\(\\le m\\), and satisfies representation stability with stability degree \\(\\le 2m\\).\n\\end{cor}\n\\begin{proof}\nThis follows from \\cite[Thm 1.10]{GL3} and \\thref{charpoly}.\n\\end{proof}\n\n\\subsection{Tensor products and \\(\\FI_G\\)-algebras}\nHere we proceed to generalize the notions introduced in \\cite[\\S4.2]{CEF} from \\(\\FI\\) to \\(\\FI_G\\). \n\nGiven \\(\\FI_G\\)-modules \\(V\\) and \\(V'\\), their tensor product \\(V \\otimes V'\\) is the \\(\\FI_G\\)-module with \\((V \\otimes V')_n = V_n \\otimes V'_n\\), where \\(\\FI_G\\) acts diagonally. \n\nA \\emph{graded \\(\\FI_G\\)-module} is a functor from \\(\\FI_G\\) to graded modules, so that each piece is graded, and the induced maps respect the grading. If \\(V\\) is graded, each graded piece \\(V^i\\) is thus an \\(\\FI_G\\)-module. If \\(V\\) and \\(W\\) are graded, the tensor product \\(V \\otimes W\\) is graded in the usual way. Say that \\(V\\) is of \\emph{finitely-generated type} if each \\(V^i\\) is finitely generated. Say that \\(V\\) is of \\emph{finite type} if it is of finitely-generated type and furthermore each \\(V^i_n\\) is finite-dimensional. Notice if \\(G\\) is a finite group, these two notions coincide.\n\nSimilarly, an \\(\\FI_G\\)-algebra is a functor from \\(\\FI_G\\) to \\(k\\)-algebras, which can also be graded. Usually our algebras will be associative, but in order to deal with Lie algebras without having to invoke anything as fancy as Poincar\\'e-Birkhoff-Witt, we do not assume associativity here. We can also define graded co-\\(\\FI_G\\)-modules and algebras, as functors from \\(\\FI_G^{\\text{op}}\\). A (co-)\\(\\FI_G\\)-algebra \\(A\\) is \\emph{generated (as an \\(\\FI_G\\)-algebra)} by a submodule \\(V\\) when each \\(A_n\\) is generated as an algebra by \\(V_n\\). \n\nFinally, there is another type of tensor product that we will need. Suppose \\(V\\) is graded \\(G\\)-module with \\(V^0 = k\\). Then the space \\(V^{\\otimes \\bullet}\\) defined by \\((V^{\\otimes \\bullet})_n = V^{\\otimes n}\\) has the structure of an \\(\\FI_G\\sharp\\)-module, as in \\cite[Defn 4.2.5]{CEF}, with the morphisms permuting and acting on the tensor factors.\n\nThe following theorem characterizes the above constructions.\n\\begin{thm} \\thlabel{alg} Let \\(G\\) be any group.\\begin{enumerate}\n\\item Let \\(V\\) and \\(V'\\) be \\(\\FI_G\\)-modules generated in degree \\(m\\) and \\(m'\\). Then \\(V \\otimes V'\\) is generated in degree \\(m + m'\\). If \\(V\\) is finitely generated and \\(V'\\) is finite type, then \\(V \\otimes V'\\) is finitely generated.\n\n\\item Let \\(A\\) be a graded (co-)\\(\\FI_G\\)-algebra generated by a graded submodule \\(V\\), where \\(V^0 = 0\\) and \\(V\\) is generated as an \\(\\FI_G\\)-module in degree \\(m\\). Then the \\(i\\)-th grades piece \\(A^i\\) is generated in degree \\(m \\cdot i\\). If \\(V\\) is of finite type, then \\(A\\) is of finite type.\n\n\\item If \\(V\\) is a graded \\(G\\)-module with \\(V^0 = k\\), then \\(V^{\\otimes \\bullet}\\) is an \\(\\FI_G\\sharp\\)-module whose \\(i\\)-th graded piece is generated in degree \\(i\\). If \\(V\\) is of finite type, then \\(V^{\\otimes \\bullet}\\) is of finite type. \n\n\\item If \\(X\\) is a connected \\(G\\)-space, then \\(H^*(X^{\\bullet}; k)\\) is an \\(\\FI_G\\sharp\\)-algebra whose \\(i\\)-th graded piece is generated in degree \\(i\\). If \\(H^*(X; k)\\) is of finite type, then \\(H^*(X^{\\bullet}; k)\\) is of finite type.\n\\end{enumerate}\n\\end{thm}\n\\begin{proof} \\ \\begin{enumerate}\n\\item This was proved by Sam-Snowden \\cite[Prop 3.1.6]{SS2} for \\(G\\) finite, and their proof applies verbatim even when \\(G\\) is infinite to show that \\(V \\otimes V'\\) is always generated in degree \\(m + m'\\), though not always finitely.\n\nFor the second part, for each \\(k \\le m + m'\\), we know \\((V \\otimes V')_k = V_k \\otimes V'_k\\) is a finitely-generated \\(W_k\\)-module, since the tensor product of a finitely-generated \\(W_k\\)-module and a finite-dimensional module is finitely generated. The result follows.\n\n\\item This was proved for \\(G\\) trivial in \\cite[Thm 4.2.3]{CEF} by appealing to the free nonassociative algebra \\(k\\{V\\}\\) generated by a vector space, and their proof applies verbatim even when \\(G\\) is infinite to show that \\(A^i\\) is always generated in degree \\(m \\cdot i\\), though not always finitely. The second part follows from the second part of (a).\n\n\\item This was proved for \\(G\\) trivial in \\cite[Prop 4.2.7]{CEF}, but here we have simplified the assumptions, by having \\(V\\) just be a single \\(G\\)-module rather than a whole \\(\\FI_G\\)-module, and so the proof is simpler. To wit, the \\(i\\)-th graded piece is\n\\[ (V^{\\otimes n})^i = \\bigoplus_{k_1 + \\cdots + k_n = i} V^{k_1} \\otimes \\cdots \\otimes V^{k_n} \\]\nAt most \\(i\\) of the nonnegative integers \\(k_1, \\dots, k_n\\) can be nonzero. Let \\(k_{j_1}, \\dots, k_{j_l}\\) be the subsequence of nonzero integers, so \\(l \\le i\\). Then for any pure tensor\n\\[ v = v_1 \\otimes \\cdots \\otimes v_n \\in V^{k_1} \\otimes \\cdots \\otimes V^{k_n},\\]\nsince \\(V^0 = k\\), we can take\n\\[ w = \\left(\\prod_{l \\notin \\{k_j\\}} v_l\\right) v_{k_{j_1}} \\otimes \\cdots v_{k_{j_l}} \\]\nand then the inclusion \\(j: [l] \\hookrightarrow [n]\\) clearly induces \\(j_* w = v\\). So by linearity, \\((V^{\\otimes n})^i\\) is generated in degree \\(i\\).\n\nIf each \\(V^i\\) is finite-dimensional, then each \\(V^{k_1} \\otimes \\cdots \\otimes V^{k_l}\\) is finite-dimensional for \\(l = 1, \\dots, i\\). Therefore \\((V^{\\otimes n})^i\\) is an \\(\\FI_G\\)-module of finite type.\n\n\n\\item This was proved for \\(G\\) trivial in \\cite[Prop 6.1.2]{CEF}, and their proof applies verbatim. It essentially follows from (3) and the K\\\"unneth formula. As \\cite{CEF} explain, technically sometimes a sign is introduced in permuting the order of tensor factors, but this does not change the proof of (3). The degree 0 part is \\(k\\) by connectivity.\n\\end{enumerate}\n\\end{proof}\n\n\\section{Orbit configuration spaces}\nLet \\(M\\) be a manifold with a free and properly discontinuous action of a group \\(G\\), so that \\(M \\to M\/G\\) is a cover. Define the \\emph{orbit configuration space} by:\n\\[\\Conf^G_n(M) = \\{(m_i) \\in M^n \\mid G m_i \\cap G m_j = \\emptyset \\text{ for } i \\ne j\\}\\]\nThis was first considered by Xicot\\'encatl in \\cite{Xi} and later investigated in e.g. \\cite{Co}, \\cite{FZ}, \\cite{CX}, \\cite{CCX}. There is a covering map \\(\\Conf^G_n(M) \\to \\Conf_n(M\/G)\\) with deck group \\(G^n\\), given by \\((m_i) \\mapsto (G m_i)\\). Thus another way to think of \\(\\Conf^G_n(M)\\) is as the space of configurations in \\(M\\) that do not degenerate upon projection to \\(M\/G\\), or as configurations in \\(M\/G\\) which also keep track of a lift in \\(M\\) of each point in the configuration.\n\nIf \\(N\\) is a normal subgroup of \\(G\\), there is an intermediate cover\n\\[\\Conf^G_n(M) \\to \\Conf^{G\/N}_n(M\/N) \\to \\Conf_n(M\/G) \\]\nwhere the first map has deck group \\(N^n\\), and the second has deck group \\((G\/N)^n\\). Also, notice that if \\(G\\) is finite, there is an embedding\n\\[ \\Conf^G_n(M) \\hookrightarrow \\Conf_{|G|n}(M), \\;\\; (m_1, \\dots m_n) \\mapsto (g_1 m_1, g_2 m_1, \\dots, g_{|G|} m_n) \\]\n\n\\subsection{\\(\\FI_G\\)-structure and finite generation for finite groups}\nFor any \\(G\\) acting discretely and properly discontinuously on \\(M\\), if we write \\(\\left(\\Conf^G(M)\\right)_n = (\\Conf^G_n(M))\\), then \\(\\Conf^G(M)\\) is a co-\\(\\FI_G\\)-space: given \\(a: [m] \\hookrightarrow [n]\\) and \\((g_i) \\in G^m\\), there is a map\n\\begin{align*}\n(a, g)^*: \\Conf^G_n(M) &\\to \\Conf^G_m(M) \\\\\n(m_i) &\\mapsto \\left(g_i m_{a(i)}\\right)\n\\end{align*}\nIn particular, if \\(G\\) is trivial we recover the usual ordered configuration space, and if \\(M = \\mathbb{C}^*\\) and \\(G = \\mathbb{Z}\/2\\mathbb{Z}\\) acting as multiplication by \\(-1\\), we obtain the type BC hyperplane complement from \\cite{Wi}. Composing with the contravariant cohomology functor, we see that \\(H^*(\\Conf^G(M), k)\\) has the structure of an \\(\\FI_G\\)-module over the field \\(k\\).\n\nWe are therefore interested in orbit configuration spaces for which \\(G\\) is virtually polycyclic, so that we can apply the results on \\(\\FI_G\\)-modules from \\S2. Interesting examples include:\n\\begin{itemize}\n\\item \\(G = \\mathbb{Z}\/2\\) acting antipodally on \\(M = S^m\\), so that \\(M\/G = \\mathbb{R}P^m\\). This was analyzed by Feichtner and Ziegler in \\cite{FZ}. Their computation of the cohomology in Thm 17 shows that \\(\\dim H^i(\\Conf^G_n(S^m); \\mathbb{Q})\\) is bounded by a polynomial in \\(n\\). We strengthen this by proving that it is in fact equal to a polynomial for \\(n \\gg 0\\). Furthermore, the \\(G^n\\) action on \\(H^i(\\Conf^G_n(S^m))\\) was analyzed in \\cite{GGSX}: in particular, their Prop 6.6 is a sort of weak form of representation stability.\n\\item \\(G = \\mathbb{Z}\/2\\) acting by a hyperelliptic involution on a 4-punctured \\(\\Sigma_g\\), with quotient a 4-punctured sphere\n\\item Any finite cover \\(\\Sigma_h \\to \\Sigma_g\\) (so that \\(h = |G|(g - 1) + 1\\)), with \\(G\\) the deck group of the covering.\n\\item \\(M = \\mathbb{R}^2\\), with \\(G\\) a lattice isomorphic to \\(\\mathbb{Z}^2\\), and \\(M\/G\\) a torus\n\\item \\(M = \\Conf_d(\\mathbb{C})\\) for some fixed \\(d\\), with \\(G = S_d\\), so that we are looking at the iterated configuration space \\(\\Conf^{S_d}_n(\\Conf_d(\\mathbb{C}))\\) and its quotient \\(\\UConf_n(\\UConf_d(\\mathbb{C}))\\). \n\\item \\(M = S^3\\), with \\(G\\) any finite subgroup of \\(\\SO(4)\\), and \\(M\/G\\) a spherical 3-manifold\n\\item Baues \\cite{Bau} proved that every torsion-free virtually polycyclic group \\(G\\) acts discretely, properly discontinuously, and cocompactly on \\(\\mathbb{R}^d\\) for some \\(d\\), and furthermore, the quotient spaces \\(\\mathbb{R}^d\/G\\) precisely comprise the \\emph{infra-solvmanifolds}. Thus for any such \\(G\\), consider \\(\\Conf^G_n(\\mathbb{R}^d)\\).\n\\end{itemize}\n\nA straightforward transversality argument (e.g., \\cite[Thm 1]{Bi}) shows that if \\(\\dim M \\ge 3\\), the map \\(\\Conf_n(M) \\hookrightarrow M^n\\) induces an isomorphism on \\(\\pi_1\\). Therefore \\(\\Conf^{\\pi_1(M)}_n(\\widetilde{M})\\) is in fact the universal cover of \\(\\Conf_n(M)\\) in dimension \\(\\ge 3\\), which provides further motivation as to why orbit configuration spaces are natural to study. Note that this does not make the last example trivial (i.e., contractible), since if \\(\\dim M > 2\\), \\(\\Conf_n(M)\\) need not be aspherical: its homotopy groups only agree with \\(M^n\\) up to \\(\\dim(M)-2\\).\n\nWe first consider the case where \\(G\\) is finite.\n\n\\begin{thm}[\\bfseries Cohomology of orbit configuration spaces] \\thlabel{confg_fg} Let \\(k\\) be a field, let \\(M\\) be a connected, orientable manifold of dimension at least 2 with each \\(H^i(M; k)\\) finite-dimensional, and let \\(G\\) be a finite group acting freely on \\(M\\). Then the \\(\\FI_G\\)-algebra \\(H^*(\\Conf^G(M); k)\\) is of finite type. \\end{thm}\n\nFollowing Church-Ellenberg-Farb-Nagpal \\cite{CEFN}, we could adapt \\thref{confg_fg} to handle \\(\\mathbb{Z}\\) coefficients. As \\cite{CEFN} mention, the proof with \\(\\mathbb{Z}\\) coefficients is essentially identical to the one with field coefficients: the difference is that one of the inputs, the analogue of our \\thref{alg}.4, becomes harder to prove. However, their proof of this analogue, \\cite[Lemm. 4.1]{CEFN}, is readily adaptable to our context. We do not make use of this except in \\S3.4.\n\n\\begin{proof}\nThe proof is based on a modification of Totaro's \\cite[Thm 1]{To}. Following Totaro, consider the Leray spectral sequence associated to the inclusion \\(\\iota: \\Conf^G_n(M) \\hookrightarrow M^n\\). This spectral sequence has the form\n\\begin{equation}\\label{tot_ss}\nH^i(M^n; R^j \\iota_* k) \\implies H^{i+j}(\\Conf^G_n(M); k)\n\\end{equation}\nwhere \\(R^j \\iota_* k\\) is the sheaf on \\(M^n\\) associated to the presheaf\n\\[U \\mapsto H^j(U \\cap \\Conf^G_n(M); k)\\]\n\nAs in Totaro's proof, the sheaf \\(R^j \\iota_* k\\) vanishes outside the appropriate ``fat diagonal'', which in this case is the union of the subspaces \\(\\Delta_{a,g,b} = \\{(m_i) \\in M^n \\mid m_a = g \\cdot m_b\\}\\), for \\(1 \\le a < b \\le n\\) and \\(g \\in G\\). Consider a point in the fat diagonal,\n\\[x = (x_1, g_{11} x_1, \\dots g_{1i_1} x_1, \\dots x_s, g_{s1} x_s, \\dots g_{s i_s} x_s).\\]\n Since \\(G\\) acts properly discontinuously, take a neighborhood of each \\(x_j\\) small enough to be disjoint from all translates of all the other neighborhoods. Then use a Riemannian metric to identify each of these with the tangent space \\(T_{x_j} X\\), \\(T_{g_{j 1} x_j} X\\), etc. \\(dg_{j k}(x_j)\\) is then an isomorphism from \\(T_{x_j} X\\) to \\(T_{g_{jk} x_j} X\\), and the condition that a point \\(m_1\\) near \\(x\\) and \\(m_2\\) near \\(g x\\) satisfy \\(m_2 = g m_1\\) becomes, upon passing to the tangent space and under the isomorphism \\(dg\\), simply the condition that \\((v,w) \\in (T_x X)^2\\) satisfy \\(v = w\\). Thus, for a neighborhood \\(U\\) of \\(x\\) small enough so that the inverse exponential map is a diffeomorphism,\n\\[\n(R^j \\iota_* k)_x = H^j(U \\cap \\Conf^G_n(M); k) = H^j(\\Conf_{i_1}(T_{x_1} X) \\times \\cdots \\times \\Conf_{i_s}(T_{x_s} X))\n\\]\nThus, the local picture looks exactly the same as in \\(\\Conf_n\\), which it should since there is a covering map \\(\\Conf_n^G(M) \\to \\Conf_n(M\/G)\\). So as in Totaro, we get generators of \\(\\Conf_n(\\mathbb{R}^d)\\), where \\(\\dim M = d\\). However, for \\(\\Conf_n(M)\\), we got just one copy of each generator \\(e_{ab}\\), coming from the diagonal \\(\\{m_a = m_b\\}\\). Here, however, we get a generator \\(e_{a,g,b}\\) for each \\(g \\in G\\), coming from each \\(\\Delta_{a,g,b}\\). The permutation action of \\(W_n\\) on the \\(\\{\\Delta_{a,g,b}\\}\\) induces an action on the \\(\\{e_{a,g,b}\\}\\), which is given by\n\\begin{equation} \\label{action}\n(\\sigma, \\vec h) \\cdot e_{a,g,b} = e_{\\sigma(a), h_a g h_b^{-1}, \\sigma(b)}\n\\end{equation}\nAs in Totaro, we can explicitly write down the relations that these \\(e_{a,g,b}\\) satisfy:\n\\begin{gather} \\begin{split} \\label{presentation}\ne_{a,g,b} &= (-1)^d e_{b,g^{-1},a} \\\\\ne_{a,g,b}^2 &= 0 \\\\\ne_{a,g,b} \\wedge e_{b,h,c} &= (e_{a,g,b} - e_{b,h,c}) \\wedge e_{a,gh,c}\n\\end{split} \\end{gather}\nTo conclude, we use the argument from \\cite[Thm 6.2.1]{CEF}. To wit, because the Leray spectral sequence is functorial, all of the spectral sequences of \\(\\Conf_n^G(M) \\hookrightarrow M^n\\), for each \\(n\\), collected together form a spectral sequence of \\(\\FI_G\\)-modules. As we just described, the \\(E_2\\) page is generated by \\(H^*(M^n; k)\\) and\nthe \\(\\FI_G\\)-module spanned by the \\(e_{a,g,b}\\). This latter is evidently finitely-generated, since it is just generated in degree 2 by \\(e_{1,e,2}\\), and the former is of finite type by \\thref{alg}.4, so therefore the \\(E_2\\) page as a whole is of finite type. The \\(E_\\infty\\) page is a subquotient of \\(E_2\\), so by Noetherianity it is of finite type, and therefore \\(H^*(\\Conf^G(M); k)\\) is of finite type.\n\\end{proof}\nWe pause briefly to dwell on the permutation action (\\ref{action}) of \\(W_n\\) on the module \\(V_n\\) spanned by \\(\\{e_{a,g,b}\\}\\), since it will come up repeatedly. Let \\(k[G]\\) be the representation of \\(G \\times G\\) where the first factor of \\(G\\) acts by multiplication on the left, and the second by multiplication on the right by the inverse (two commuting left actions). Therefore\n\\[V_n = \\Ind_{W_2 \\times W_{n-2}}^{W_n} k[G] \\otimes k = \\Ind^{\\FI_G}(k[G])_n \\]\n\nRecall that, as a \\((k[G], k[G]^{\\text{op}})\\)-bimodule, the regular representation has the following decomposition into irreducibles:\n\\[ k[G] = \\bigoplus_{\\chi \\in \\Irr(G)} V_{\\chi} \\boxtimes (V_{\\chi})^* \\]\nThus if we turn this into a \\(k[G] \\otimes k[G]\\)-module by having the right factor act by \\(g^{-1}\\), this becomes\n\\[k[G] = \\bigoplus_{\\chi \\in \\Irr(G)} V_{\\chi} \\boxtimes V_{\\chi} = \\bigoplus_{\\chi \\in \\Irr(G)} L( (2)_\\chi) \\]\nas \\(k[G] \\otimes k[G] = k[G \\times G]\\)-modules. Therefore\n\\begin{equation} \\label{irr_decom}\nV = \\bigoplus_{\\chi \\in \\Irr(G)} \\Ind^{\\FI_G}((2)_{\\chi}).\n\\end{equation}\nas an \\(\\FI_G\\)-module, so it is manifestly an \\(\\FI_G\\sharp\\)-module. In particular, if \\(G\\) is trivial, we obtain \\(\\Ind^{\\FI}((2)) = \\Sym^2 k^n \/ k^n\\), consistent with the computation done in \\cite{CEF}.\n\n\\begin{cor}\nLet \\(M\\) be a connected, orientable manifold of dimension at least 2 with each \\(H^i(M; \\mathbb{Q})\\) finite-dimensional, let \\(G\\) be a finite group acting freely on \\(M\\), and let \\(k\\) be a splitting field for \\(G\\) of characteristic 0. Then for each \\(i\\), the characters of the \\(W_n\\)-representations \\(H^i(\\Conf^G_n(M); k)\\) are given by a single character polynomial for all \\(n \\gg 0\\).\n\\end{cor}\n\n\n\\thref{confg_fg} has another consequence, which as far as we can tell is a new result (recall that \\cite[Thm 6.2.1]{CEF} only applied to orientable manifolds).\n\n\\begin{cor}\nLet \\(M\\) be a connected, non-orientable manifold of dimension at least \\(2\\) with \\(H^*(M; \\mathbb{Q})\\) of finite type. Then the \\(\\FI\\)-algebra \\(H^*(\\Conf(M); \\mathbb{Q})\\) is of finite type.\n\\end{cor}\n\\begin{proof}\nConsider the orientation cover \\(\\widetilde{M} \\to M\\), which has deck group \\(G = \\mathbb{Z}\/2\\). Since there is a covering map \\(\\Conf^G_n(\\widetilde{M}) \\to \\Conf_n(M)\\), with deck group \\(G^n\\), then by transfer there is an isomorphism\n\\[H^*(\\Conf_n(M); \\mathbb{Q}) \\cong \\left(H^*(\\Conf^G_n(\\widetilde{M}); \\mathbb{Q})\\right)^{S_n}\\]\nBy \\thref{confg_fg}, \\(H^*(\\Conf^G_n(\\widetilde{M}); \\mathbb{Q})\\) is a finite type \\(\\FI_G\\)-algebra, and therefore \\(H^*(\\Conf_n(M); \\mathbb{Q})\\) is a finite type FI-algebra.\n\\end{proof}\n\nHomological stability for non-orientable manifolds, which is a consequence of this corollary, was proven by Randal-Williams in \\cite{RW}.\n\n\\subsection{Dealing with infinite groups}\nWe pause to explain the complications that arise when \\(G\\) is infinite, before proceeding to at least partially resolve them. As a toy example, forgetting for a moment the setting of \\(\\FI_G\\) and sequences of spaces, consider the space \\(X = S^1 \\vee S^1 = K(F_2, 1)\\), with one loop called \\(a\\) and the other \\(b\\). Let \\(Y\\) be the \\(G := \\mathbb{Z}\\) cover associated to the kernel of the map \\(F_2 \\to \\mathbb{Z}\\), \\(a \\mapsto 0, b \\mapsto 1\\). Hence \\(Y\\) is an infinite sequence of line segments labeled \\(b\\) joining loops labeled \\(a\\). So \\(Y\\) is homotopy equivalent to a wedge of infinitely many circles, and thus \\(H_1(Y; k)\\) has infinite rank. However, notice that the covering group \\(G\\) acts on \\(Y\\), and that \\(H_1(Y; k) \\cong k[G] = k[b^{\\pm 1}]\\) as \\(G\\)-modules.\n\nIn particular, \\(H_1(Y; k)\\) is finitely-generated as a \\(G\\)-module. However, looking at cohomology, \\(H^1(Y; k) = \\Hom_k(H_1(Y), k) = \\Hom_k(k[G], k) = k^G\\). Thus, \\(H_1(Y; k)\\) consists of finite linear combinations of elements of \\(G\\), and \\(H^1(Y; k)\\), infinite linear combinations. In particular, \\(H^1(Y; k)\\) is no longer finitely generated as a \\(G\\)-module. However, notice that it contains a dense submodule isomorphic to \\(H_1(Y; k)\\).\n\nNow we see why the proof of \\thref{confg_fg} does not suffice when \\(G\\) is infinite: in general, the \\(E_2\\) page is not just be generated by the \\(e_{a,g,b}\\), which is to say, by finite linear combinations of them, but instead it is generated by all infinite linear combinations of them. Therefore the \\(E_2\\) page is in general not a finitely-generated \\(\\FI_G\\)-module. \n\nIf we are willing to settle for ``presented in finite degree''---for example, if \\(G\\) is not virtually polycyclic---then this is good enough:\n\\begin{thm} \\thlabel{confg_coh}\nLet \\(M\\) be a connected, orientable manifold of dimension at least 2. Then the \\(\\FI_G\\)-algebra \\(H^*(\\Conf^G_n(M); k)\\) is presented in finite degree.\n\\end{thm}\n\\begin{proof}\nThe argument from \\thref{confg_fg} carries over essentially directly. The \\(E_2\\) page of the spectral sequence (\\ref{tot_ss}) is generated as a \\(k\\)-algebra by \\(H^*(M^n; k)\\) and the dual space to \\(\\langle e_{a,g,b} \\rangle\\), that is, the space of infinite linear combinations\n\\[ \\sum_{a,g,b} v_{a,g,b} e_{a,g,b} = \\sum_{1 \\le a < b \\le n} \\left( \\sum_{g \\in G} v_{a,g,b} e_{a,g,b} \\right) \\]\nThis space is evidently generated as an \\(\\FI\\)-module by those sums of the form \\(\\sum_{g \\in G} v_{g} e_{1,g,2}\\), since we can get all other \\(a,b\\) by appropriate permutations. These generators evidently live in degree 2, and the relations (\\ref{presentation}) all live in degree 3, so that the \\(E_2\\) page is presented in finite degree. By \\thref{coh}, taking successive pages in the spectral sequences preserves being presented in finite degree, as does passing from the \\(E_\\infty\\) page to the final cohomology. So we conclude that \\(H^*(\\Conf^G_n(M); k)\\) is presented in finite degree.\n\\end{proof}\n\nHowever, if we want to preserve finite generation, the analysis at the beginning of this subsection suggests that the correct thing to look at is actually \\(H_\\bullet(\\Conf^G(M); k)\\) instead. Unfortunately, on the face of it, \\(H_\\bullet(\\Conf^G(M); k)\\) is a co-\\(\\FI_G\\)-module, so since the maps go ``in the wrong way'', it is never finitely generated.\n\nHowever, when the quotient \\(M\/G\\) is an \\emph{open} manifold, we obtain the following generalization of \\cite[Prop 6.4.2]{CEF}.\n\\begin{thm}[\\bfseries Orbit configuration spaces of open manifolds]\n\\thlabel{boundary_sharp}\nLet \\(N\\) be the interior of a connected, compact manifold \\(\\overline{N}\\) of dimension \\(\\ge 2\\) with nonempty boundary \\(\\partial\\overline{N}\\), and let \\(\\pi: M \\to N\\) be a \\(G\\)-cover, so that \\(G\\) acts freely and properly discontinuously on \\(M\\). Then \\(\\Conf^G(M)\\) has the structure of a homotopy \\(\\FI_G\\sharp\\)-space, that is, a functor from \\(\\FI_G\\sharp\\) to hTop, the category of spaces and homotopy classes of maps.\n\\end{thm}\n\\begin{proof}\nWe follow the argument in \\cite{CEF}. Fix a collar neighborhood \\(S\\) of one component of \\(\\partial \\overline{N}\\), let \\(R = \\pi^{-1}(S)\\) and let \\(R_0\\) be a connected component of \\(R\\), and fix a homeomorphism \\(\\Phi: M \\cong M \\setminus \\overline{R}\\) isotopic to the identity (\\(\\Phi\\) and the isotopy both exist by lifting). For any inclusion of finite sets \\(X \\subset Y\\), define a map\n\\[ \\Psi^Y_X: \\Conf^G_X(M) \\to \\Conf^G_Y(M)\\]\nup to homotopy, as follows. First, if \\(Y = X\\), set \\(\\Psi^Y_X = \\id\\). Next, note that a configuration in \\(\\Conf^G_X(M)\\) is just an embedding \\(X \\hookrightarrow M\\) (that stays injective upon composition with \\(\\pi\\)). So fix an element \\(q_X^Y: (Y - X) \\hookrightarrow R_0\\) of \\(\\Conf^G_{Y - X}(R_0)\\). Then any embedding \\(f: X \\hookrightarrow M\\) in \\(\\Conf^G_X(M)\\) extends to an embedding \\(\\Psi^Y_X(f): Y \\hookrightarrow M\\) by\n\\[ \\Psi^Y_X(f)(t) = \\begin{cases} \\Phi(f(t)) & t \\in X \\\\ q^Y_X(t) & t \\notin X \\end{cases} \\]\nThe image of \\(\\pi \\circ \\Phi\\) is disjoint from \\(S\\), while the image of \\(\\pi \\circ q^Y_X\\) is contained in \\(S\\), so the above map never send points in \\(X\\) into the same \\(G\\)-orbit as it sends points outside of \\(X\\), and therefore this does give an element of \\(\\Conf^G_Y(M)\\). Furthermore, since \\(\\Conf^G_{Y - X}(R_0)\\) is connected (since \\(R_0\\) is, and \\(\\dim R_0 \\ge 2\\)), different choices of \\(q^Y_X\\) give homotopic maps, so \\(\\Phi^Y_X\\) is well-defined up to homotopy.\n\nNow, an \\(\\FI_G\\sharp\\) morphism \\(Z \\to Y\\) consists of an injection \\(X \\hookrightarrow Z\\), an injection \\(X \\hookrightarrow Y\\), and a tuple \\(g: X \\to G\\). Normally if we were extending from an \\(\\FI_G\\) structure, we would think of \\(X\\) as being a subset of \\(Z\\), but since we are extending from a \\emph{co}-\\(\\FI_G\\) structure, it is more natural to think of \\(X\\) as a subset of \\(Y\\), with an explicit map \\(a: X \\to Z\\). The induced map is then given by\n\\begin{align*}\n\\Conf^G_Z(M) & \\to \\Conf^G_X(M) \\xrightarrow{\\Psi^Y_X} \\Conf^Y(M) \\\\\n(m_i) & \\mapsto (g_i m_{a(i)})\n\\end{align*}\nIt is straightforward to verify that this is functorial up to homotopy, as \\cite[Prop 6.4.2]{CEF} do for trivial \\(G\\).\n\\end{proof}\nIn particular, when the conditions of \\thref{boundary_sharp} hold, then \\(H_*(\\Conf^G(M); k)\\) is an \\(\\FI_G\\sharp\\)-module. We want to argue that, when \\(G\\) is virtually polycyclic, \\(H_*(\\Conf^G(M); k)\\) is finitely-generated type. To do this, we need the following.\n\\begin{prop} \\thlabel{arnold}\nLet \\(A(G,d)\\) be the \\(\\FI_G\\sharp\\)-algebra where \\(A(G)_n\\) has generators \\(\\{e_{a,g,b} \\mid 1 \\le a \\ne b \\le n, g \\in G\\}\\) of degree \\(d-1\\) and action given by (\\ref{action}), modulo the following relations:\n\\begin{align*}\ne_{a,g,b} &= (-1)^d e_{b,g^{-1},a} \\\\\ne_{a,g,b}^2 &= 0 \\\\\ne_{a,g,b} \\wedge e_{b,h,c} &= (e_{a,g,b} - e_{b,h,c}) \\wedge e_{a,gh,c}\n\\end{align*}\nThen \\(A(G,d)\\) is an \\(\\FI_G\\sharp\\)-module of finitely-generated type.\n\\end{prop}\n\\begin{proof}\nFirst, by construction \\(A(G,d)\\) is presented in finite degree, so it remains to show that each \\(A(G,d)_n\\) is of finitely-generated type. For convenience, put \\(D = d-1\\), so that \\(A(G,d)_n\\) is only nonzero is degree divisible by \\(D\\). It is straightforward to verify (e.g., see [ARNOLD]) that \\(A(G,d)^{iD}_n\\) is spanned as a vector space by all products of the form\n\\begin{equation} \\label{arnold_span}\nv = e_{a_1, g_1, b_1} \\wedge e_{a_2, g_2, b_2} \\wedge \\cdots \\wedge e_{a_i, g_i, b_i} \\;\\; \\text{ where } a_s < b_s, \\;\\;\\; b_1 < b_2 < \\cdots < b_i, \\;\\;\\; g_s \\in G\n\\end{equation}\nWe will describe an explicit procedure which, given such a \\(v\\), finds an element \\(\\vec h \\in G^n\\) so that\n\\[\nv' := \\vec h \\cdot v = e_{a_1, e, b_1} \\wedge e_{a_2, e, b_2} \\wedge \\cdots \\wedge e_{a_i, e, b_i}\n\\]\nthat is, so that each \\(g'_i = e\\) in (\\ref{arnold_span}). We construct \\(\\vec h\\) inductively. To begin, we put a copy of \\(g_1\\) in the \\(b_1\\)-th coordinate of \\(\\vec h\\), which cancels out the \\(g_1\\) in (\\ref{arnold_span}). We multiply \\(v\\) by this partial \\(\\vec h\\), which may have the effect of modifying later \\(g_i\\)'s. We then proceed and put a copy of (the new, modified) \\(g_2\\) in the \\(b_2\\)-th coordinate of \\(\\vec h\\), which cancels the \\(g_2\\) in (\\ref{arnold_span}). We can do this with no trouble because we know \\(b_1 < b_2\\). Again, we use this to modify \\(v\\) and proceed to \\(b_3\\), etc. Eventually we have constructed our \\(\\vec h\\) and modified \\(v\\) so that each \\(g_s = e\\).\n\nWe therefore conclude that \\(A(G,d)^{iD}_n\\) is finitely generated as a \\(G^n\\)-module, so \\emph{a fortiori} as a \\(W_n\\)-module. Therefore \\(A(G,d)\\) is of finitely-generated type.\n\\end{proof}\n\nWe therefore obtain the following.\n\\begin{thm}[\\bfseries Homology of orbit configuration spaces]\n\\thlabel{confg_fgsharp}\nLet \\(N\\) be the interior of a connected, compact manifold of dimension \\(\\ge 2\\) with nonempty boundary. Let \\(M \\to N\\) be a \\(G\\)-cover, with \\(G\\) virtually polycyclic, such that \\(H_*(M)\\) is of finite type. Then \\(H_*(\\Conf^G(M); k)\\) is a finitely-generated type \\(\\FI_G\\sharp\\)-module.\n\\end{thm}\n\\begin{proof}\nWe still need to appeal to cohomology, in order to make use of the cup product structure. So the proof follows that of \\thref{confg_fg}, but only considering the sub-\\(\\FI_G\\)-module of the \\(E_2\\) page that actually is generated by \\(H^*(M^n)\\) and the \\(e_{a,g,b}\\), and not the infinite linear combinations of them. Notice that this is isomorphic to the \\(E^2\\) page associated to the appropriate spectral sequence computing the homology of \\(\\Conf^G(M)\\) (to be technical, this comes from \\emph{cosheaf} homology). \n\nThis submodule of the \\(E_2\\) page is precisely the algebra described in \\thref{arnold}. Thus it is of finitely-generated type, and therefore the \\(E^2\\) page for homology is an \\(\\FI_G\\)-module of finitely-generated type. The same final argument from \\thref{confg_fg} (which uses Noetheriantiy, since \\(G\\) is virtually polycyclic) thus shows that \\(H_*(\\Conf^G(M); k)\\) is of finitely-generated type.\n\\end{proof}\n\n\\begin{cor}\nLet \\(N\\) be the interior of a connected, compact manifold of dimension \\(\\ge 2\\) with nonempty boundary, let \\(M \\to N\\) be a \\(G\\)-cover, with \\(G\\) virtually polycyclic, \\(M\\) connected and orientable and \\(H^*(M; \\mathbb{Q})\\) of finite type. Then for each \\(i\\), the \\(W_n\\)-representations \\(H_i(\\Conf^G_n(M); \\mathbb{Q})\\) satisfy \\(K_0\\)-stability.\n\\end{cor}\n\n\\subsection{Homotopy groups of configuration spaces}\nThe main application of the theory of \\(\\FI_G\\)-modules until this point has been in Kupers-Miller's \\cite{KM} work on the homotopy groups of configuration spaces. They prove that, for \\(M\\) a simply-connected manifold of dimension at least 3, the dual homotopy groups \\(\\Hom(\\pi_i(\\Conf_n(M)), \\mathbb{Z})\\) form a finitely-generated \\(\\FI\\)-module. In \\cite[\\S5.2]{KM}, they sketch an extension of this result to the non-simply connected case. Kupers-Miller are naturally led to consider \\(\\Conf_n^{\\pi_1(M)}(\\widetilde{M})\\), since as we have said it is the universal cover of \\(\\Conf_n(M)\\) once \\(\\dim M \\ge 3\\), and so has the same higher homotopy groups as \\(\\Conf_n(M)\\).\n\nOur results on orbit configuration spaces are able to confirm most of Kupers-Miller's sketch, while also clarifying some oversights. As stated, their Prop 5.8 is not correct, since as we have seen, if \\(G\\) is infinite, in general we cannot conclude that \\emph{co}homology is finitely generated. Instead, the best we can do is our \\thref{confg_fg} and \\thref{confg_fgsharp}, where we either assume that \\(G\\) is finite or that \\(M\\) is an open manifold. We therefore obtain the following. Note that, following Kupers-Miller, we work here with \\(\\mathbb{Z}\\) coefficients, since as we mentioned, we could rework \\thref{confg_fg} to use \\(\\mathbb{Z}\\) coefficients.\n\n\\begin{thm}[\\bfseries Homotopy groups of configuration spaces, take 1]\nLet \\(M\\) be a connected manifold of dimension \\(\\ge 3\\) with finite fundamental group \\(G\\) such that \\(H^*(M)\\) is finite-dimensional. For \\(i \\ge 2\\), the dual homotopy groups \\(\\Hom(\\pi_i(\\Conf_n(M)), \\mathbb{Z})\\) and \\(\\Ext^1_{\\mathbb{Z}}(\\pi_i(\\Conf_n(M)), \\mathbb{Z})\\) are finitely generated \\(\\FI_G\\)-modules.\n\nIn particular, if \\(k\\) is a splitting field for \\(G\\) of characteristic 0, then for each \\(i\\), the characters of the \\(W_n\\)-representations \\(\\Hom(\\pi_1(\\Conf_n(M)), k)\\) are given by a single character polynomial for \\(n \\gg 0\\), and \\(\\{\\Hom(\\pi_1(\\Conf_n(M)), k)\\}\\) satisfies representation stability.\n\\end{thm}\n\nApplying \\thref{confg_coh}, we obtain the following.\n\\begin{thm}[\\bfseries Homotopy groups of configuration spaces, take 2]\nLet \\(M\\) be a connected, orientable manifold of dimension at least 2. Then the dual homotopy groups \\(\\Hom(\\pi_i(\\Conf_n(M), \\mathbb{Z})\\) and \\(\\Ext^1_{\\mathbb{Z}}(\\pi_i(\\Conf_n(M)), \\mathbb{Z})\\) are \\(\\FI_G\\)-modules presented in finite degree.\n\\end{thm}\n\nApplying \\thref{confg_fgsharp}, we also obtain the following.\n\n\\begin{thm}[\\bfseries Homotopy groups of configuration spaces, take 3]\nLet \\(M\\) be the interior of a connected, compact manifold with nonempty boundary of dimension \\(\\ge 3\\) such that \\(G = \\pi_1(M)\\) is virtually polycyclic. For \\(i\\) at least 2, \\(\\pi_i(\\Conf_n(M))\\) is a finitely-generated \\(\\FI_G\\sharp\\)-module. In particular, \\(\\pi_i(\\Conf_n(M)) \\otimes \\mathbb{Q}\\) satisfies \\(K_0\\)-stability.\n\n\\end{thm}\n\n\\subsection{Intermediate configuration spaces and \\(\\FI \\times G\\)}\nNow suppose that \\(G\\) is abelian, and consider the following map:\n\\[\\psi: G^n \\rtimes S_n \\to G \\times S_n,\\;\\; \\psi(\\vec{g}, \\sigma) = (g_1 \\cdots g_n, \\sigma) \\]\nSince \\(G\\) is abelian and \\(\\sigma\\) leaves the product of all the \\(g_i\\)'s invariant, \\(\\psi\\) is a homomorphism. Its kernel is the subgroup \\(\\ker( \\vec{g} \\mapsto g_1 \\cdots g_n) \\subset G^n\\), which as a group is just isomorphic to \\(G^{n-1}\\). Since \\(G\\) is a \\(\\mathbb{Z}\\)-module, we can identify \\(\\ker \\psi\\) as \\(G \\otimes_\\mathbb{Z} V\\), where \\(V\\) is the standard representation of \\(S_n\\) of rank \\(n-1\\). We obtain the following diagram of covering spaces refining \\(\\Conf^G_n(M) \\twoheadrightarrow \\UConf_n(M\/G)\\):\n\\begin{equation*}\n\\begin{diagram}\n&&\\Conf^G_n(M)&& \\\\\n&&\\dTo_{G^{n-1}}&&\\\\\n&& \\Conf'_n(M\/G)&& \\\\\n&\\ldTo^{S_n} && \\rdTo^{G} & \\\\\n\\UConf'_n(M\/G) &&&& \\Conf_n(M\/G) \\\\\n&\\rdTo_{G} && \\ldTo_{S_n} & \\\\\n&&\\UConf_n(M\/G)&&\n\\end{diagram}\n\\end{equation*}\nwhere \\(\\Conf'_n(M\/G)\\) and \\(\\UConf'_n(M\/G)\\) are the intermediate covers. These do not have particularly nice descriptions in general, but if \\(M\\) is a Lie group and \\(G\\) a discrete central subgroup, so that \\(M\/G\\) is also a Lie group, then\n\\[ \\Conf'_n(M\/G) = \\{(m_1 \\dots m_n; m) \\in (M\/G)^n \\times M\\,|\\,m_i \\ne m_j\\,\\forall i\\ne j;\\, \\pi(m) = m_1 \\cdots m_n\\} \\]\nand \\(\\UConf'_n(M\/G)\\) is its quotient by \\(S_n\\). So \\(\\Conf'\\) parameterizes configurations of points in the base space with a chosen lift of the \\emph{product} of the points. When \\(M\/G = \\mathbb{C}^*\\), these spaces are closely related to the Burau representation, as we shall see.\n\nDefine \\(\\FI \\times G\\) to be the usual product of categories, where we think of \\(G\\) as a category with one object. An \\(\\FI \\times G\\)-module is thus a sequence of \\(G\\)-representations equipped with the structure of an \\(\\FI\\)-module, where the two actions commute. The restriction functor \\(\\Res: \\FI \\times G \\to G \\times S_n\\) still has a left adjoint \\(\\Ind^{\\FI \\times G}: G \\times S_n \\to \\FI \\times G\\) given by\n\\[ \\Ind^{\\FI \\times G}(V)_{n+k} = \\Ind^{S_{n+k}}_{S_n \\times S_k} V \\]\nWe can likewise define the category \\(\\FI\\sharp \\times G\\). Then \\(\\Ind^{\\FI \\times G}(V)\\) naturally have the structure of an \\(\\FI\\sharp \\times G\\)-module, and the analogue of \\thref{sharp_decomp} shows that any \\(\\FI\\sharp \\times G\\)-module is a direct sum of these.\n\nNotice that \\(H^*(\\Conf'_n(M\/G); \\mathbb{Q})\\) is naturally an \\(\\FI \\times G\\)-module. Furthermore, if \\(M\\) satisfies the condition of \\thref{boundary_sharp} then \\(H_*(\\Conf'_n(M\/G); \\mathbb{Q})\\) is an \\(\\FI\\sharp \\times G\\)-module.\n\n\\begin{prop}\n\\thlabel{prime_fg}\nIf \\(M\\) is a connected orientable manifold of dimension at least 2 with \\(H^*(M; \\mathbb{Q})\\) of finite type, and \\(G\\) a finite abelian group acting freely and properly discontinuously on \\(M\\), then \\(H^*(\\Conf'(M\/G); \\mathbb{Q})\\) is an \\(\\FI \\times G\\)-algebra of finitely-generated type.\n\nIf \\(N\\) is the interior of a connected compact manifold of dimension \\(\\ge 2\\) with nonempty boundary, \\(M \\to N\\) is a \\(G\\)-cover with \\(G\\) f.g. abelian, \\(M\\) is connected and orientable and \\(H^*(M; \\mathbb{Q})\\) is a finite type \\(G\\)-module, then \\(H_*(\\Conf'(M\/G); \\mathbb{Q})\\) is an \\(\\FI\\sharp \\times G\\)-module of finitely-generated type.\n\\end{prop}\n\\begin{proof}\nIf \\(G\\) is finite this follows immediately by transfer applied to \\(\\Conf^G_n(M)\\). However, even if \\(G\\) is infinite, we can still apply the Serre spectral sequence to the fibration\n\\[\\Conf^G_n(M) \\to \\Conf'_n(M\/G) \\to BG^{n-1}\\]\nThis spectral sequence has the form\n\\[ E_{i,j}^2 = H_i(BG^{n-1}; H_j(\\Conf^G_n(M); \\mathbb{Q})) \\implies H_{i+j}(\\Conf'_n(M\/G); \\mathbb{Q}) \\]\nBy finite generation of \\(H_j(\\Conf^G_n(M); \\mathbb{Q})\\) as an \\(\\FI_G\\)-module, each of the terms \\(E_{i,j}^2\\) is a finitely-generated \\(\\FI \\times G\\)-module, and thus the whole \\(E^2\\) page is of finitely-generated type. So the same argument at the end of \\thref{confg_fg} thus shows that \\(H_*(\\Conf'_n(M\/G); \\mathbb{Q})\\) is an \\(\\FI \\times G\\)-module of finitely-generated type.\n\\end{proof}\n\\thref{prime_fg} thus has the following consequence.\n\\begin{cor}\n\\thlabel{gstab}\nLet \\(M \\to N\\) be a \\(G\\)-cover with \\(G\\) f.g. abelian, and \\(M\\) a connected orientable manifold of dimension \\(\\ge 2\\) with \\(H^*(M; \\mathbb{Q})\\) of finite type. Suppose further that either \\(G\\) is finite, or \\(N\\) is the interior of a compact manifold with nonempty boundary. Then for any \\(i \\ge 0\\), the sequence of \\(G\\)-modules \\(H_i(\\UConf'_n(N); \\mathbb{C})\\) are isomorphic for \\(n \\gg 0\\).\n\\end{cor}\n\\begin{proof}\nThe \\(S_n\\)-cover \\(\\Conf'_n(M\/G) \\to \\UConf'_n(M\/G)\\) gives a transfer isomorphism of \\(G\\)-modules:\n\\[H_i(\\UConf'_n(M\/G); \\mathbb{Q}) \\cong \\left(H_i(\\Conf'_n(M\/G); \\mathbb{Q})\\right)_{S_n}\\]\n\nIf \\(G\\) is finite then a finitely generated \\(\\FI \\times G\\)-module is still finitely generated when just considered an \\(\\FI\\)-module. Thus \\(H_i(\\Conf'_n(M\/G); \\mathbb{Q})\\) is a finitely generated \\(\\FI\\)-module, and representation stability tells us that \\(H_i(\\Conf'_n(N); \\mathbb{C})_{S_n}\\) stabilize when \\(n \\gg 0\\). By the transfer isomorphism, we conclude the corollary.\n\nOn the other hand, suppose \\(G\\) is infinite and \\(N\\) is the interior of a closed manifold with boundary. Then \\(H_i(\\Conf'(M\/G); \\mathbb{Q})\\) is a finitely generated \\(\\FI\\sharp \\times G\\)-module by \\thref{prime_fg}. Therefore\n\\[ H_i(\\Conf'(M\/G); \\mathbb{Q}) = \\bigoplus_j \\Ind^{\\FI \\times G}(W_j) \\]\nfor some finite collection \\(W_1, \\dots W_k\\) of representations of \\(S_{i_1} \\times G, \\dots, S_{i_k} \\times G\\). By Frobenius reciprocity,\n\\[ \\left(\\Ind^{\\FI \\times G}(W_j)_n\\right)_{S_n} \\cong (W_j)_{S_{i_j}} \\]\nThus once \\(n\\) is at least the degree of generation of \\(H_i(\\Conf'_n(M\/G); \\mathbb{Q})\\), the coinvariants \\(H_i(\\UConf'_n(M\/G); \\mathbb{Q})\\) will always be isomorphic as \\(G\\)-modules. The conclusion again follows, and in this case we can explicitly say that stability occurs once we reach the degree of generation of \\(H_i(\\Conf'_n(M\/G); \\mathbb{Q})\\). This is not necessarily the same as the degree of generation of \\(H_i(\\Conf_n(M\/G; \\mathbb{Q})\\), since the spectral sequence argument of \\thref{prime_fg} makes non-constructive use of Noetherianity.\n\\end{proof}\n\nNotice that the sequence of \\(G\\)-modules \\(H_i(\\UConf'_n(M\/G); \\mathbb{C})\\) do not have any obvious maps between them, or obvious larger category-representation structure. It is only by lifting to the cover, noticing that we have representation stability, and then descending, that we obtain actual stability of \\(G\\)-modules. In this respect, \\thref{gstab} mirrors the proof of homological stability for unordered configuration spaces of manifolds in \\cite{CEF} (this result was originally proven in \\cite{Chu}). On the other hand, the \\(\\FI_G\\)-module structure of \\(H_i(\\Conf^G(M))\\) is crucial for the specifics of \\thref{gstab}. If we forgot about the \\(\\FI_G\\)-module structure, and merely thought of \\(H^i(\\Conf^G(M); \\mathbb{C})\\) as an FI-module, finite generation as an FI-module would be enough to show that \\(\\UConf'(M\/G))\\) satisfies homological stability \\emph{in dimension} (i.e., as vector spaces), but it would not show that it satisfies homological stability \\emph{as \\(G\\)-modules}.\n\nNotice also that if \\(G\\) is infinite, then we are really getting a strong result about \\(G\\)-modules. We do not need to make things nicer by passing to the Grothendieck group: we get honest stability of \\(G\\)-modules. This despite the fact that in general we can say very little about what all the finitely-generated \\(G\\)-modules actually look like.\n\n\\section{Complex reflection groups}\nOne classical example of a finite wreath product is the \\emph{complex reflection group} \\(G(d,1,n)\\), where \\(G = \\mathbb{Z}\/d\\mathbb{Z}\\) and so \\(W_n = \\mathbb{Z}\/d\\mathbb{Z} \\wr S_n\\), also sometimes referred to as the \\emph{full monomial group}. Because of the action of \\(G\\) on \\(\\mathbb{C}\\) as multiplication by \\(d\\)-th roots of unity, \\(W_n\\) acts on \\(\\mathbb{C}^n\\) by generalized permutation matrices whose entries are \\(d\\)-th roots of unity. This gives rise to a rich class of examples of \\(\\FI_G\\)-modules, which we now describe.\n\nNote that since \\(G\\) is abelian, a splitting field for \\(G\\) is just the same as a field containing all the character values of \\(G\\). So in this case there is a minimal splitting field of characteristic zero, namely the field generated by the character values, which is \\(\\mathbb{Q}(\\zeta_d)\\). \n\n\\subsection{Orbit configuration space of complex reflection groups}\nFirst, of course, we can consider the orbit configuration space where \\(M = \\mathbb{C}^*\\) and \\(G = \\mathbb{Z}\/d\\mathbb{Z}\\) acting, as just described, as multiplication by \\(d\\)-th roots of unity. Thus\n\\[\n\\Conf^G_n(\\mathbb{C}^*) = \\left\\{(v_i) \\in \\mathbb{C}^n \\mid v_i \\ne \\zeta_d^k v_j \\text{ for } i \\ne j, v_i \\ne 0 \\text{ for all } i\\right\\}\n\\]\n\\(\\Conf^G_n(\\mathbb{C}^*)\\) is thus the complement of the hyperplanes fixed by the standard complex-reflection generators of \\(W_n\\). This arrangement, called the \\emph{complex reflection arrangement}, is much studied. For instance, Bannai \\cite{Ba} proved that the complement is aspherical; its fundamental group is referred to as the \\emph{pure monomial braid group}, and sometimes denoted \\(P(d,n)\\). Thus the cohomology of \\(\\Conf^G_n(\\mathbb{C}^*)\\) is isomorphic to the group cohomology of \\(P(d,n)\\).\n\nOrlik-Solomon \\cite{OS} calculated the cohomology of the complement of any hyperplane arrangement, as follows. Let \\(\\mathcal{A}\\) be a collection of linear hyperplanes in \\(\\mathbb{C}^n\\), and put \\(M(\\mathcal{A}) = \\mathbb{C}^n \\setminus \\bigcup \\mathcal{A}\\). Say that a subset \\(\\{H_1, \\dots, H_p\\} \\subset \\mathcal{A}\\) is \\emph{dependent} if \\(H_1 \\cap \\dots \\cap H_p = H_1 \\cap \\cdots \\widehat{H_i} \\cdots \\cap H_p\\); alternately, if the linear forms defining the \\(H_i\\) are linearly dependent. Now let\n\\[\nE(\\mathcal{A}) = \\bigwedge \\langle e_H \\mid H \\in \\mathcal{A}\\rangle = \\bigwedge H^1(M(\\mathcal{A}); \\mathbb{Q})\n\\]\nand let \\(I(\\mathcal{A}) \\subset E(\\mathcal{A})\\) be the ideal\n\\[\nI(\\mathcal{A}) = \\left( \\sum_{i = 1}^p (-1)^{i} e_{H_1} \\wedge \\cdots \\wedge \\widehat{e_{H_i}} \\wedge \\cdots \\wedge e_{H_p}\\,\\middle|\\, H_1, \\dots, H_p \\text{ are dependent}\\right).\n\\]\nThen \\(H^*(M(\\mathcal{A}); \\mathbb{Q}) = E(\\mathcal{A})\/I(\\mathcal{A})\\). We then obtain the following.\n\\begin{thm}[\\bfseries Cohomology of complex reflection arrangements]\n\\thlabel{crg_fg}\n\\(H^*(\\Conf^{\\mathbb{Z}\/d\\mathbb{Z}}_n(\\mathbb{C}^*); \\mathbb{Q}) = H^*(P(d,1,n); \\mathbb{Q})\\) is a finite type \\(\\FI_G\\sharp\\)-algebra. For each \\(i\\), the characters of the \\(W_n\\)-representations \\(H^i(P(d,1,n); \\mathbb{Q}(\\zeta_d))\\) are given by a single character polynomial of degree \\(2i\\) for all \\(n\\), and therefore \\(H^i(P(d,1,n; \\mathbb{Q}(\\zeta_d))\\) satisfies representation stability with stability degree \\(\\le 4i\\).\n\\end{thm}\n\\begin{proof}\nSince \\(\\mathbb{C}^*\\) is the interior of an compact orientable 2-manifold with boundary, \\thref{confg_fgsharp} says that \\(H^*(\\Conf^{\\mathbb{Z}\/d\\mathbb{Z}}_n(\\mathbb{C}^*); \\mathbb{Q})\\) is a finitely-generated \\(\\FI_G\\sharp\\)-module. To determine the degree this \\(\\FI_G\\)-module is generated in, we use the Orlik-Solmon presentation. The hyperplane arrangement is:\n\\[\\mathcal{A} = \\{z_i \\mid 1 \\le i \\le n\\} \\cup \\{e_{i,a,j} \\mid 1 \\le i \\ne j \\le n; a \\in G\\}\\]\nwhere \\(z_i\\) is the hyperplane \\(\\{v_i = 0\\}\\) and \\(e_{i,a,j}\\) is the hyperplane \\(\\{v_i = \\zeta^a v_j\\}\\), so that \\(e_{i,a,j} = e_{j,a^{-1},i}\\). To understand \\(I(\\mathcal{A})\\), it is helpful to consider the embedding mentioned in \\S3.1,\n\\[ \\Conf^G_n(\\mathbb{C}^*) \\hookrightarrow \\Conf_{|G|n}(\\mathbb{C}),\\;\\; (v_1, \\dots, v_n) \\mapsto (v_1, \\zeta v_1, \\zeta^2 v_1, \\dots, \\zeta^{d-1} v_n) \\]\nThis induces a map \\(H^*(\\Conf_{|G|n}(\\mathbb{C}); \\mathbb{Q}) \\to H^*(\\Conf^G_n(\\mathbb{C}^*); \\mathbb{Q})\\). Orlik-Solomon likewise determines the cohomology of \\(\\Conf_{|G|n}(\\mathbb{Q})\\). The hyperplane arrangement for \\(\\Conf_{|G|n}(\\mathbb{C})\\) is\n\\[\\mathcal{B} = \\{f_{i, g; j, h} \\mid 1 \\le i, j \\le n;\\, g, h \\in G;\\, (i, g) \\ne (j, h)\\} \\]\nThe induced map \\(H^1(\\Conf_{|G|n}(\\mathbb{C}); \\mathbb{Q}) \\to H^1(\\Conf^G_n(\\mathbb{C}^*); \\mathbb{Q})\\) is just given by\n\\[ f_{i,g; j, h} \\mapsto \\begin{cases} e_{i,h\/g,j} & \\text{if } i \\ne j \\\\ z_{i} & \\text{if } i = j \\end{cases}\\]\nand is thus evidently surjective. Since \\(H^*(\\Conf^G_n(\\mathbb{C}^*); \\mathbb{Q})\\) is generated by \\(H^1\\), this means that the total induced map \\(H^*(\\Conf_{|G|n}(\\mathbb{C}); \\mathbb{Q}) \\to H^*(\\Conf^G_n(\\mathbb{C}^*); \\mathbb{Q})\\) is surjective. We can therefore describe the ideal \\(I(\\mathcal{A})\\) in terms of the simple relations generating the ideal \\(I(\\mathcal{B})\\) of the braid arrangement. We obtain the following presentation:\n\\[\nH^*(M(\\mathcal{A}); \\mathbb{Q}) = \\bigwedge\\langle e_{i,a,j}, z_k \\rangle \\bigg\/ \\left\\langle \\begin{matrix} e_{i,a,j} \\wedge e_{j,b,k} = (e_{i,a,j} - e_{j,b,k}) \\wedge e_{i,ab,k} \\\\ z_i \\wedge z_j = (z_i - z_j) \\wedge e_{i,a,j} \\\\ e_{i,a,j} \\wedge e_{i,b,j} = (e_{i,a,j} - e_{i,b,j})\\wedge z_i \\end{matrix} \\right\\rangle\n\\]\nAnother way to view these is by writing \\(e_{i,a,i} = z_i\\) for any \\(a \\ne 1 \\in G\\), as suggested by the induced map on \\(H^1\\) above: then the first relation, if \\(i,j,k\\) are allowed to equal one another, implies the others. As described in \\S3.1, \\(G^n \\rtimes S_n\\) acts on \\(H^1\\) as follows: on the \\(\\{z_i\\}\\), \\(G^n\\) acts trivially and \\(S_n\\) acts in the standard way, and on the \\(\\{e_{i,a,j}\\}\\) the action is:\n\\[ \\left(\\sigma, \\zeta^{\\vec{b}}\\right) \\cdot e_{i,a,j} = e_{\\sigma(i),\\,a - b_i + b_j,\\,\\sigma(j)} \\]\n\nSo we see that \\(H^*(\\Conf^{\\mathbb{Z}\/d\\mathbb{Z}}_n(\\mathbb{C}^*); \\mathbb{Q})\\) is generated as an algebra by \\(H^1(\\Conf^{\\mathbb{Z}\/d\\mathbb{Z}}_n(\\mathbb{C}^*); \\mathbb{Q})\\), and that \\(H^1\\) is generated as an \\(\\FI_G\\)-module by \\(\\{e_{1,a,2}\\}\\) and \\(z_1\\), and is therefore generated in degree \\(2\\). By \\thref{alg}, \\(H^i\\) is generated in degree \\(2i\\), and so the stable range follows from \\thref{sharp_range}.\n\\end{proof}\n\nWilson \\cite[Thm 7.14]{Wi} proved the case \\(d = 2\\), for which the arrangement is the type \\(BC\\) braid arrangement. The decomposition for \\(H^1\\), following (\\ref{irr_decom}), is\n\\[ H^1(P(d,1,n); \\mathbb{Q}) = \\Ind^{\\FI_G}((1)_{\\chi_0}) \\oplus \\bigoplus_{\\chi \\in \\Irr(G)} \\Ind^{\\FI_G}((2)_{\\chi}) \\]\nwhere \\(\\chi_0\\) is the trivial character of \\(G\\). That is, \\(\\Ind^{\\FI_G}((1)_{\\chi_0})\\) picks out the submodule spanned by the \\(\\{z_i\\}\\), and \\(\\bigoplus_{\\chi \\in \\Irr(G)} \\Ind^{\\FI_G}((2)_{\\chi})\\) the submodule spanned by the \\(\\{e_{i,a,j}\\}\\). We can also compute the decomposition for \\(H^2\\):\n\\begin{gather*} \nH^2(P(d,1,n); \\mathbb{Q}) = \\Ind^{\\FI_G}( (2,1)_{\\chi_0}) \\oplus \\Ind^{\\FI_G}( (3)_{\\chi_0}) \\oplus \\bigoplus_{\\chi \\in \\Irr(G)} \\Ind^{\\FI_G}( (3,1)_\\chi) \\oplus \\Ind^{\\FI_G}( (2)_\\chi) \\\\\n\\oplus \\bigoplus_{\\substack{\\chi \\in \\Irr(G) \\\\ \\chi \\ne \\chi_0}} \\Ind^{\\FI_G}( (2)_\\chi) \\oplus \\Ind^{\\FI_G}((1)_{\\chi_0}, (2)_\\chi)^2 \\oplus \\Ind^{\\FI_G}((1)_\\chi, (1,1)_\\chi) \\oplus \\Ind^{\\FI_G}((2)_{\\chi_0}, (2)_\\chi)\n\\end{gather*}\nThese calculations agree with Wilson's \\cite[p. 123]{Wi} for the case \\(d = 2\\).\n\n\\subsection{Diagonal coinvariant algebras}\nComplex reflection groups also provide a generalization of the coinvariant algebra, as follows. Let \\(k\\) be a field with a primitive \\(d\\)-th root of unity \\(\\zeta_d\\), and let \\(V_n = k^n\\) be the canonical representation of \\(W_n\\) by \\(\\zeta_d\\)-power permutation matrices. Consider\n\\[\nk[V_n^{\\oplus r}] \\cong k[x_1^{(1)}, \\cdots, x_n^{(1)}, \\cdots, x^{(r)}_1, \\cdots x^{(r)}_n]\n\\]\nThis has a natural grading by \\(r\\)-tuples \\((j_1, \\dots, j_r)\\), where \\(j_i\\) counts the total degree in the variables \\(x_1^{(i)}, \\dots, x_n^{(i)}\\). There is an action of \\(W_n\\) induced by the diagonal action on \\(V^{\\oplus r}\\) which respects the grading. Let \\(\\mathcal{I}_n = (k[V_n]^{W_n}_+)\\) be the ideal generated by the constant-term-free \\(W_n\\)-invariant polynomials. The \\emph{diagonal coinvariant algebra} is\n\\[\\mathcal{C}^{(r)}(n) = k[V_n^{\\oplus r}] \/ \\mathcal{I}_n\\]\nThese \\(\\mathcal{C}^{(r)}(n)\\) arise naturally because, by the celebrated Chevalley-Shephard-Todd Theorem, complex reflection groups exactly comprise the class of groups \\(G\\) acting on a vector space \\(V\\) for which \\(k[V]^G\\) is a polynomial algebra. The representation theory of \\(\\mathcal{C}^{(r)}(n)\\) has been much studied. The case \\(r = 1\\) is classical, going back to Chevalley's original paper \\cite{Ch} proving the general case of Chevalley-Shephard-Todd.\n\nThe case \\(r = 2\\) for a complex reflection group (\\(d > 2\\)) was first studied by Haimain \\cite{Ha} and Gordon \\cite{Go}. The case of a general \\(r\\) was analyzed by Bergeron in \\cite{Ber}. Bergeron shows that for a fixed \\(n\\), the multigraded Hilbert polynomial of \\(\\mathcal{C}^{(r)}(n)\\) can be described independently of \\(r\\). (In a sense, this is orthogonal to our work, where \\(r\\) is fixed and \\(n\\) varies.) In general, very little is known about the \\(\\mathcal{C}^{(r)}(n)\\) for \\(n \\ge 3\\), even their dimension.\n\\begin{thm}[\\bfseries Finite generation for diagonal coinvariant algebras]\nThe sequence of coinvariant algebras \\(\\mathcal{C}^{(r)}(n)\\) form a graded co-\\(\\FI_{\\mathbb{Z}\/d\\mathbb{Z}}\\)-algebra of finite type.\n\\end{thm}\n\\cite[Thm 1.11]{CEF} proved the case \\(d = 1\\), and Wilson \\cite[Thm 7.8]{Wi} proved it for \\(d = 2\\).\n\\begin{proof}\nLet \\(V\\) be the \\(\\FI_G\\)-module associated to the \\(n\\)-dimensional \\(W_n\\)-representations \\(V_n = k^n\\), where \\(G\\) acts by multiplication by \\(\\zeta_d\\). Let \\(\\chi_1\\) be the character of \\(\\mathbb{Z}\/d\\mathbb{Z}\\) sending 1 to \\(\\zeta_d\\), and let \\(\\underline{\\lambda}\\) be the \\(G\\)-partition of 1 supported on \\(\\chi_1\\), so \\(\\underline{\\lambda}(\\chi_1) = (1)\\). Then \\(V_n = L(\\underline{\\lambda})_n \\oplus L(\\underline{\\lambda})\\). Thus \\(k[V^{\\oplus r}]\\) is naturally a co-\\(\\FI_G\\)-module of finite type. The ideals \\(\\mathcal{I}_n\\) together form a co-\\(\\FI_G\\)-ideal \\(\\mathcal{I}\\), determined by the \\(W_n\\) action and the maps\n\\begin{align*}\n(\\iota_n)^*: \\mathcal{I}_{n+1} &\\to \\mathcal{I}_n \\\\\nx_i &\\mapsto \\begin{cases} x_i & i \\le n \\\\ 0 & i = n+1 \\end{cases}\n\\end{align*}\nThus \\(\\mathcal{C}^{(r)}\\) is a co-\\(\\FI_G\\)-quotient of a co-\\(\\FI_G\\)-algebra of finite type, so it is a co-\\(\\FI_G\\)-algebra of finite type. Since \\(W_n\\) preserves the grading of \\(\\mathcal{C}^{(r)}(n)\\), then as a co-\\(\\FI_G\\)-module, \\(\\mathcal{C}^{(r)}\\) decomposes into graded pieces \\(\\mathcal{C}^{(r)}_J\\).\n\\end{proof}\nBy Noetherianity, each graded piece \\(\\mathcal{C}^{(r)}_J\\) is a finitely-generated co-\\(\\FI_G\\)-module by \\thref{alg}. So we obtain the following.\n\\begin{cor}\nFor each multi-index \\(J\\), the characters of the \\(W_n\\)-representations \\(\\mathcal{C}^{(r)}_J(n)\\) are given by a single character polynomial for all \\(n \\gg 0\\). \n\\end{cor}\n\n\n\\subsection{Lie algebras associated to the lower central series}\nFor any group \\(\\Gamma\\), recall that the \\emph{lower central series} is defined inductively by \\(\\Gamma_1 = \\Gamma\\) and \\(\\Gamma_{k+1} = [\\Gamma_k, \\Gamma]\\). The \\emph{associated graded Lie algebra} \\(\\gr(\\Gamma)\\) over a field \\(k\\) is defined as\n\\[\n\\gr(\\Gamma) = \\bigoplus_{k \\ge 1} (\\Gamma_{k+1}\/\\Gamma_k) \\otimes_{\\mathbb{Z}} k\n\\]\nwhere the Lie algebra structure is induced by the commutator. For example, if \\(\\Gamma\\) is a free group of rank \\(n\\), then \\(\\gr(\\Gamma)\\) is the free lie algebra of rank \\(n\\). Since the higher \\(\\Gamma_i\\) are generated by iterated commutators, \\(\\gr(\\Gamma)\\) is always generated as a Lie algebra by \\(\\gr(\\Gamma)_1 = H_1(\\Gamma;k)\\).\n\nNow, as \\cite{CEF} carefully note, \\(\\pi_1\\) is not a functor from topological spaces to groups, since it depends on a basepoint, and a map of spaces only determines a group homomorphism up to conjugacy. However, since conjugation acts trivially on the associated Lie algebra, there is a functor \\(Z \\mapsto \\gr \\pi_1(Z)\\) from the category Top of topological spaces to the category of graded Lie algebras. Thus \\(\\gr \\pi_1(\\Conf^G(M))\\) has the structure of a graded co-\\(\\FI_G\\)-algebra. \n\nHere we continue to look at the case \\(M = \\mathbb{C}^*\\), \\(G = \\mathbb{Z}\/d\\), so \\(\\pi_1 = P(d,n)\\). The associated graded Lie algebra \\(\\gr P(d,n)\\) has been studied, for example, by D. Cohen in \\cite{Co}, who obtained some partial results, but still, little is known about it.\n\\begin{thm}[\\bfseries Finite generation for \\(\\gr P(d,n)\\)]\nFor each \\(i\\), the characters of the \\(W_n\\)-representations \\(\\gr P(d,n)^i\\) are given by a single character polynomial for all \\(n\\).\n\\end{thm}\n\\begin{proof}\n\\(H_1(\\Conf^G(\\mathbb{C}^*)\\) is a finitely generated \\(\\FI_G\\sharp\\)-module by \\thref{crg_fg}. Since \\(H_1(\\Gamma)\\) generates \\(\\gr(\\Gamma)\\) as a Lie algebra, this means that \\(\\gr P(d,n)\\) is an \\(\\FI_G\\sharp\\)-algebra of finite type by \\thref{alg}. Thus by Noetherianity, its graded pieces are finitely generated \\(\\FI_G\\sharp\\)-modules.\n\\end{proof}\n\n\\cite[Thm 7.3.4]{CEF} proved the case \\(d = 1\\), for which \\(P(1,n) = P_n\\), the pure braid group.\n\n\\section{Affine arrangements}\nA motivating example of an \\(\\FI_G\\)-module for \\(G\\) infinite, and one with interesting applications to affine braid groups, is the analogue of \\(\\Conf^{\\mathbb{Z}\/d}(\\mathbb{C}^*)\\) when we ``let \\(d\\) go to infinity'' and replace \\(\\mathbb{Z}\/d\\) with the infinite group \\(\\mathbb{Z}\\).\nWhereas the \\(\\mathbb{Z}\/d\\)-cover of \\(\\mathbb{C}^*\\) is just homeomorphic to \\(\\mathbb{C}^*\\) again, the \\(\\mathbb{Z}\\)-cover of \\(\\mathbb{C}^*\\) is homeomorphic to \\(\\mathbb{C}\\). To be specific, we take \\(M = \\mathbb{C}\\) and \\(G = \\langle \\tau \\rangle \\cong \\mathbb{Z}\\) acting as translation by some \\(\\tau \\in \\mathbb{C}\\). Thus \\(M\/G\\) is homeomorphic to a cyclinder, which is homeomorphic to the punctured plane, and the quotient map is \\(\\exp(x\/2\\pi i \\tau): \\mathbb{C} \\to \\mathbb{C}^*\\).\n\nAs in the case of complex reflection groups, \\(\\Conf_n^G(M)\\) is the complement of the hyperplane arrangement\n\\[ \\mathcal{A}_n = \\{(z_i) \\in \\mathbb{C}^n \\mid z_i - z_j \\not\\in \\tau\\mathbb{Z}\\}\\]\nso one might hope that Orlik-Solomon gives the homology or cohomology. Unfortunately, \\(\\mathcal{A}_n\\) is an infinite arrangement, and their arguments hinge on inducting on the number of hyperplanes (by looking at the ``deleted'' and ``restricted'' arrangements), so their theorem is not directly applicable.\n\nHowever, a result of Cohen-Xicot\\'encatl \\cite{CX} provides a solution. They compute the homology and cohomology of a similar space---that of \\(\\Conf^{\\mathbb{Z}^2}(\\mathbb{C})\\), mentioned earlier, so that the quotient \\(\\mathbb{C}\/\\mathbb{Z}^2\\) is a torus---using a Fadell-Neuwirth type argument, that inducts on the dimension of the space, rather than the number of hyperplanes. \n\nIn fact, the argument they give proves the following general result. Following Falk-Randell \\cite{FR}, we first make the following definitions. Let \\(\\mathcal{A}\\) be an arrangement of hyperplanes in \\(\\mathbb{C}^n\\), and put \\(M = \\mathbb{C}^n \\setminus \\bigcup \\mathcal{A}\\). Say that \\(M\\) is \\emph{strictly linearly fibered} if after a suitable linear change of coordinates the restriction to the first \\(n-1\\) coordinates is a fiber bundle projection with base \\(N\\) the complement of an arrangement in \\(\\mathbb{C}^{n-1}\\) and fiber a complex line with a discrete set of points removed. Now define a \\emph{fiber-type} arrangement inductively as follows: \\begin{itemize}\n\\item The 1-arrangement \\(\\{0\\} \\subset \\mathbb{C}\\) is \\emph{fiber-type}\n\\item An \\(n\\)-arrangement for \\(n \\ge 2\\) is \\emph{fiber-type} if the complement \\(M\\) is strictly linearly fibered over \\(N\\), and \\(N\\) is the complement of an \\((l-1)\\)-arrangement of fiber type.\n\\end{itemize}\nThus the complement \\(M\\) of a fiber-type \\(n\\)-arrangement sits atop a tower of fibrations\n\\begin{equation} \\label{fiber-type}\nM = M_n \\xrightarrow{p_n} M_{n-1} \\xrightarrow{p_{n-1}} \\cdots \\to M_2 \\xrightarrow{p_2} M_1 = \\mathbb{C}^*\n\\end{equation}\nwith the fiber \\(F_k\\) of \\(p_k\\) homeomorphic to \\(\\mathbb{C}\\) with a discrete set of points removed.\n\nIn particular, Falk-Randell use the tower of fibrations (\\ref{fiber-type}) to prove that \\(M\\) is aspherical, that each bundle map \\(p_k\\) has a section, and that \\(\\pi_1(M_{k-1})\\) acts trivially on \\(H_i(F_k)\\).\n\n\n\\begin{prop}[{\\cite[Prop 6]{CX}}]\nLet \\(\\mathcal{A}\\) be an arrangement of hyperplanes in \\(\\mathbb{C}^n\\) (possibly countably infinite and affine) which is discrete in the space of all hyperplanes. Put \\(M = \\mathbb{C}^n \\setminus \\bigcup \\mathcal{A}\\), and suppose \\(\\mathcal{A}\\) is fiber-type.\n\nThen \\(H^*(M; \\mathbb{Q})\\) contains a dense subalgebra \\(A\\) generated by classes \\(E_{H} \\in H^1(M; \\mathbb{Q})\\) for each \\(H \\in \\mathcal{A}\\), which is isomorphic to the Orlik-Solomon algebra of \\(\\mathcal{A}\\). Moreover, there is an isomorphism \\(H_i(M; \\mathbb{Q}) \\cong A^i\\).\n\\end{prop}\n\nCohen-Xicot\\'encatl use the tower of fibrations (\\ref{fiber-type}) to obtain this result. In a sense, this result was basically implicit in Falk-Randell \\cite{FR}, but there the arrangements were always assumed to be finite.\n\nThe algebra \\(A\\) is the algebra one gets by following the recipe of the Orlik-Solomon algebra of \\(\\mathcal{A}\\), even though \\(\\mathcal{A}\\) happens to be infinite, with each generator \\(E_H\\) coming from monodromy around the hyperplane \\(H\\). So the situation is really as good as one could hope for.\n\n\\(\\Conf^\\mathbb{Z}(\\mathbb{C})\\) is closely related to affine Coxeter and Artin groups. The affine Coxeter group \\(W_{\\widetilde{A}_n}\\) is the one associated to the Coxeter diagram \\(\\widetilde{A}_n\\), which is obtained by adding one extra node connecting the first and last nodes to the Coxeter diagram \\(A_n\\). Concretely, let \\(V^\\mathbb{R} = \\{(x_i) \\in \\mathbb{R}^n \\mid \\sum x_i = 0\\}\\). Then \\(W_{\\widetilde{A}_{n-1}}\\) is the group generated by reflections in \\(V^\\mathbb{R}\\) through the hyperplanes \\(H^\\mathbb{R}_s = \\{x_i - x_j = n \\mid n \\in \\mathbb{Z}\\}\\). This group \\(W_{\\widetilde{A}_{n-1}}\\) is isomorphic to \\(\\mathbb{Z}^{n-1} \\rtimes S_n\\). Let \\(V\\) be the complexification of \\(V^\\mathbb{R}\\), and let \\(V_0 = V \\setminus \\bigcup H_s\\) the hyperplane complement. Then \\(\\pi_1(V_0)\\) is the pure Artin braid group \\(P_{\\widetilde{A}_{n-1}}\\), the group \\(W_{\\widetilde{A}_{n-1}}\\) acts freely on \\(V_0\\) and \\(\\pi_1(V_0\/W_{\\widetilde{A}_{n-1}})\\) is the Artin braid group \\(B_{\\widetilde{A}_{n-1}}\\). But notice that \\(V_0\\) is just the fiber of the bundle \\(\\Conf^\\mathbb{Z}(\\mathbb{C}) \\to \\mathbb{C}\\) given by taking the sum of the coordinates, which is trivial since \\(\\mathbb{C}\\) is contractible, so that \\(V_0 \\simeq \\Conf^\\mathbb{Z}(\\mathbb{C})\\). In fact, it has been proven \\cite{Ok} that \\(V_0\\) is aspherical, and therefore \\(\\Conf^\\mathbb{Z}(\\mathbb{C})\\) is a \\(K(P_{\\widetilde{A}_{n-1}}, 1)\\).\n\nWe then obtain the following.\n\n\\begin{thm}[\\bfseries Homology of type \\(\\widetilde{A}_n\\) pure braid group]\n\\(H_*(\\Conf^{\\mathbb{Z}}(\\mathbb{C}); \\mathbb{Q}) = H_*(P_{\\widetilde{A}_{n-1}}; \\mathbb{Q})\\) is a finitely-generated type \\(\\FI_G \\sharp\\)-module. For each \\(i\\), the \\(W_n\\)-representations \\(H_i(P_{\\widetilde{A}_{n-1}}; \\mathbb{Q})\\) satisfies \\(K_0\\)-stability with stability degree \\(\\le 4i\\).\n\\end{thm}\n\\begin{proof}\nBy \\thref{confg_fgsharp}, \\(H_*(\\Conf^\\mathbb{Z}(\\mathbb{C}); \\mathbb{Q})\\) is a finitely-generated type \\(\\FI_G\\sharp\\)-module, while \\(H^*(\\Conf^\\mathbb{Z}(\\mathbb{C}); \\mathbb{Q})\\) is an \\(\\FI_G\\sharp\\) algebra not of finitely-generated type.\n\nTaking the usual coordinates on \\(\\mathbb{C}^n\\), we see that \\(\\Conf^\\mathbb{Z}(\\mathbb{C})\\) is a fiber-type arrangement. Proposition 5.1 therefore applies, and we obtain a presentation for \\(H_*(\\Conf^\\mathbb{Z}(\\mathbb{C}); \\mathbb{Q})\\).\n\nRecall \\cite{OT} that for affine arrangements, the Orlik-Solomon ideal is generated not just by \\(\\partial e_S\\) for \\(S\\) dependent, but also by \\(e_S\\) when \\(\\cap S = \\emptyset\\) (possible for non-central arrangements), which is derived by considering the homogeneized arrangement. In the case of \\(\\Conf^\\mathbb{Z}(\\mathbb{C})\\), \\(\\mathcal{A}\\) consists of the affine hyperplanes \\(H_{i,a,j} = \\{z_i - z_j = a \\mid a \\in \\mathbb{Z}\\}\\), and thus\n\\[\nA = \\bigwedge\\langle e_{i,a,j} \\rangle \\bigg\/ \\left\\langle \\begin{matrix} e_{i,a,j} \\wedge e_{j,b,k} = (e_{i,a,j} - e_{j,b,k}) \\wedge e_{i,a+b,k} \\\\ e_{i,a,j} \\wedge e_{i,b,j} = 0 \\end{matrix} \\right\\rangle\n\\]\nand \\(H_i(\\Conf^\\mathbb{Z}(\\mathbb{C}); \\mathbb{Q}) \\cong A^i\\). As an algebra, \\(A\\) is generated by \\(A^1\\), which is generated as an \\(\\FI_G\\)-module in degree 2, and so by \\thref{alg}, \\(A^i\\) is generated in degree \\(2i\\). So by Theorem 2.5, \\(H_i(\\Conf^\\mathbb{Z}(\\mathbb{C}); \\mathbb{Q})\\) satisfies \\(K_0\\)-stability with stability degree \\(4i\\).\n\n\\end{proof}\nFurthermore, since the Coxeter group \\(W_{\\widetilde{A}_{n-1}} \\cong \\mathbb{Z}^{n-1} \\rtimes S_n\\), then \\(V_0 \/ W \\simeq \\UConf'_n(\\mathbb{C}^*)\\), in the notation of \\S3.1, and therefore \\(\\UConf'_n(\\mathbb{C}^*)\\) is a classifying space for the full Artin group \\(B_{\\widetilde{A}_{n-1}}\\). Here, Callegaro-Moroni-Salvetti \\cite[Thm 4.2]{CMS} computed that\n\\begin{align*}\nH_i(B_{\\widetilde{A}_{n-1}}; \\mathbb{Q}) = \\begin{cases} \\mathbb{Q}[t^{\\pm 1}]\/(1 + t) & 1 \\le i \\le n - 2 \\\\ \\mathbb{Q}[t^{\\pm 1}]\/(1 + t) & \\text{if } i = n-1, n \\text{ odd} \\\\ \\mathbb{Q}[t^{\\pm 1}]\/(1 - t^2) & \\text{if } i = n-1, n \\text{ even} \\end{cases}\n\\end{align*}\nas \\(\\mathbb{Q}[G] = \\mathbb{Q}[t^{\\pm 1}]\\)-modules. This verifies the stabilization as \\(\\mathbb{Q}[G]\\)-modules for \\(n \\gg 0\\) proven in Corolarry 3.5, and in fact stabilization occurs once \\(n \\ge i + 2\\).\n\nThese spaces also have close connections to the Burau representation. First, if we let\n\\[\\UConf_{n, 1}(\\mathbb{C}) = \\{(\\{x_1, \\dots, x_n\\}; x) \\mid x_i \\ne x_j, x_i \\ne x\\}\\] then there is a bundle \n\\[\\UConf_n(\\mathbb{C}^*) \\to \\UConf_{n,1}(\\mathbb{C}) \\to \\mathbb{C}.\\]\nSince \\(\\mathbb{C}\\) is contractible, \\(\\UConf_n(\\mathbb{C}^*) \\simeq \\UConf_{n,1}(\\mathbb{C})\\). Similarly, let\n\\[\\UConf'_n(\\mathbb{C}^*) = \\{(\\{x_1, \\dots, x_n\\}; y) \\mid x_i \\ne 0, x_i \\ne x_j, e^y = x_1 \\cdots x_n\\}\\]\nand let\n\\[\\UConf'_{n, 1} = \\{(\\{x_1, \\dots, x_n\\}; x, y) \\mid x_i \\ne x, x_i \\ne x_j, e^y = (x_1 - x) \\cdots (x_n - x)\\}.\\]\nThen \\(\\UConf'_n(\\mathbb{C}^*) \\cong \\UConf'_{n, 1}\\). Finally, notice that the inclusion \\(\\UConf_n(\\mathbb{C}^*) \\hookrightarrow \\UConf_n(\\mathbb{C})\\) is homotopy equivalent to the bundle projection \\(\\UConf(\\mathbb{C})_{n,1} \\twoheadrightarrow \\UConf(\\mathbb{C})\\) onto the first \\(n\\) coordinates. We therefore obtain the following diagram:\n\\begin{equation*}\n\\begin{diagram}\n\\UConf'_n(\\mathbb{C}^*) && \\simeq && \\UConf'_{n,1}(\\mathbb{C}) \\\\\n\\dTo &&&& \\dTo \\\\\n\\UConf_n(\\mathbb{C}^*) && \\simeq && \\UConf_{n,1}(\\mathbb{C}) \\\\\n& \\rdTo && \\ldTo & \\\\\n&& \\UConf_n(\\mathbb{C}) &&\n\\end{diagram}\n\\end{equation*}\nThus \\(\\pi_1(\\UConf_n(\\mathbb{C})) = B_n\\) acts up to conjugacy on the homotopy fiber of the map from \\(\\UConf'_n(C^*)\\), which is just the actual fiber of the map from \\(\\UConf'_{n,1}(\\mathbb{C})\\). Call this fiber \\(X_f\\) for \\(f \\in \\UConf_n(\\mathbb{C})\\). This gives an honest action of \\(B_n\\) on \\(H_1(X_f; \\mathbb{C})\\), which is the Burau representation. Furthermore, replacing the group \\(G = \\mathbb{Z}\\) with \\(\\mathbb{Z}\/d\\mathbb{Z}\\), this construction gives the Burau representation specialized at \\(\\zeta_d = e^{2\\pi i\/d}\\). Chen has explored this connection and its relation to point-counts of superelliptic curves in \\cite{WCh}; in his notation, with \\(G = \\mathbb{Z}\/d\\mathbb{Z}\\), \\(\\UConf'_{n,1}(\\mathbb{C})\\) is called \\(E_{n,d}\\). \n\nFinally, we likewise obtain results on the pure braid group of type \\(\\widetilde{C}_n\\).\n\\begin{thm}[\\bfseries Homology of the type \\(\\widetilde{C}_n\\) pure braid group]\n\\(H_*(P_{\\widetilde{C}_n}; \\mathbb{Q})\\) is a finitely-generated type \\(\\FI_G\\) module, for \\(G = \\mathbb{Z} \\rtimes \\mathbb{Z}\/2\\). For each \\(i\\), the \\(W_n\\)-representation \\(H_i(P_{\\widetilde{C}_n}; \\mathbb{Q})\\) satisfies \\(K_0\\)-stability with stability degree \\(\\le 4i\\).\n\\end{thm}\n\\begin{proof}\nAs \\cite[\\S4.3]{Mor} explains, the hyperplane complement for type \\(\\widetilde{C}_n\\) is the following:\n\\[ Y_n = \\{x \\in \\mathbb{C}^n \\mid x_i \\pm x_j \\notin \\mathbb{Z}, x_i \\notin \\tfrac{1}{2}\\mathbb{Z}\\} \\]\nApplying the map \\(z \\mapsto \\exp(2\\pi i z)\\) componentwise gives a normal covering map \\(Y_n \\to Y'_n\\) with deck group \\(\\mathbb{Z}^n\\), where\n\\[ Y'_n = \\{x \\in (\\mathbb{C}^*)^n \\mid x_i \\ne x_j^{\\pm 1}, x_i \\ne \\pm 1\\} \\]\nThe map \\(z \\mapsto \\frac{1-z}{1+z}\\) is a self-homeomorphism of \\(\\mathbb{C} \\setminus \\{0,1,-1\\}\\) taking \\(1\/z\\) to \\(-z\\). Applying this map componentwise to \\(Y'_n\\) therefore gives a homeomorphism\n\\[ Y'_n \\cong \\{x \\in \\mathbb{C}^n \\mid x_i \\ne \\pm x_j, x_i \\ne 0, 1, -1\\} \\]\nFinally, applying the map \\(z \\mapsto z^2\\) componentwise gives a normal covering map \\(Y'_n \\to \\Conf_n(\\mathbb{C} \\setminus \\{0,1\\})\\), with deck group \\((\\mathbb{Z}\/2)^n\\). In conclusion, if we write \\(M = \\mathbb{C} \\setminus \\frac{1}{2}\\mathbb{Z}\\), there is a normal cover \\(M \\to \\mathbb{C} \\setminus \\{0,1\\}\\) given by \\(z \\mapsto \\left( \\dfrac{1 - e^{2\\pi i z}}{1 + e^{2\\pi i z}}\\right)^2\\) with deck group \\(G = \\mathbb{Z} \\rtimes \\mathbb{Z}\/2\\), and \\(Y_n\\) is homeomorphic to the orbit configuration space \\(\\Conf^G_n(M)\\). \n\nBy \\thref{confg_fgsharp}, \\(H_*(\\Conf^G(M); \\mathbb{Q})\\) is a finitely-generated type \\(\\FI_G\\sharp\\)-module. Again, we use Prop 5.1 to obtain a presentation. The hyperplane arrangement consists of the hyperplanes \\(E_{i,a,j} = \\{z_i - z_j = a \\mid a \\in \\mathbb{Z}\\}\\), \\(H_{i,a,j} = \\{z_i + z_j = a \\mid a \\in \\mathbb{Z}\\}\\), and \\(Z_{i,a} = \\{2z_i \\ne a \\mid a \\in \\mathbb{Z}\\}\\). The Orlik-Solomon algebra is given by:\n\\[ A = \\bigwedge \\langle e_{i,a,j}, h_{i,a,j}, z_{i,b} \\rangle \\bigg\/ \\left\\langle \\begin{matrix} e_{i,a,j} \\wedge e_{j,b,k} = (e_{i,a,j} - e_{j,b,k}) \\wedge e_{i,a+b,k}; \\;\\; e_{i,a,j} \\wedge e_{i,b,j} = 0 \\\\ \ne_{i,a,j} \\wedge h_{j,b,k} = (e_{i,a,j} - h_{j,b,k}) \\wedge h_{i,a+b,k}; \\; \\; h_{i,a,j} \\wedge h_{i,b,j} = 0 \\\\\ne_{i,a,j} \\wedge h_{i,b,j} = (e_{i,a,j} - h_{i,b,j}) \\wedge z_{i, a+b}; \\; \\; z_{i,a} \\wedge z_{i,b} = 0 \\\\\nz_{i,a} \\wedge z_{j,b} = (z_{i,a} - z_{j,a}) \\wedge h_{i, \\frac{a+b}{2}, j} \\text{ if } 2 \\mid a + b, 0 \\text{ otherwise}\n\\end{matrix} \\right\\rangle \\]\nand \\(H_i(\\Conf^G(M); \\mathbb{Q}) \\cong A^i\\). In particular, we see that this \\(\\FI_G\\sharp\\)-module is generated in degree \\(2i\\). So by Theorem 2.5, \\(H_i(\\Conf^G(M); \\mathbb{Q})\\) satisfies \\(K_0\\)-stability with stability degree \\(4i\\).\n\n\\end{proof}\nIn particular, we can again use the calculation (\\ref{irr_decom}) to write \\(H_1\\) as an induced module:\n\\begin{equation} \\label{affC}\nH_1(P_{\\widetilde{C}_{n-1}}; \\mathbb{Q}) = \\Ind_1^{\\FI_{G}} k[\\mathbb{Z}] \\oplus \\Ind_2^{\\FI_{G}} (V \\oplus W)\n\\end{equation}\nwhere \\(V\\) and \\(W\\) are both isomorphic as vector spaces to \\(k[\\mathbb{Z}]\\), but as representations of \\(k[\\mathbb{Z} \\times \\mathbb{Z}]\\), \\(V\\) has one copy of \\(\\mathbb{Z}\\) acting positively and one acting negatively, whereas \\(W\\) has both acting positively. In this case, the image of \\(V\\) and \\(W\\). In this case, \\([V] = [W] = 0\\) in \\(K_0(\\mathbb{Z} \\times \\mathbb{Z})\\), so we as expected we lose a lot of information in passing to \\(K_0\\). However, the first summand in (\\ref{affC}) is retained in \\(K_0\\). To wit, the representation \\(U = k[\\mathbb{Z}]\\) of \\(\\mathbb{Z} \\rtimes \\mathbb{Z}\\) is just \\(\\Ind_{\\mathbb{Z}\/2}^{\\mathbb{Z} \\rtimes \\mathbb{Z}\/2} k\\), so by Moody's induction theorem, this is nonzero in \\(K_0(\\mathbb{Z} \\rtimes \\mathbb{Z}\/2)\\). We therefore obtain, as we saw in the introduction,\n\\[ [H_1(P_{\\widetilde{A}_{n-1}}; \\mathbb{Q})] = [\\Ind_{W_1 \\times W_{n-1}}^{W_n} U] = [L((n)_U)] \\]\n\n\\section{Automorphism groups of free products}\nGiven a group \\(G\\), let \\(G^{*n} = G \\ast \\cdots \\ast G\\) be the \\(n\\)-fold free product. The group \\(\\Aut(G^{*n})\\) contains a copy of \\(\\Aut(G) \\wr S_n\\), acting by automorphisms on each factor and permuting them. This normalizes the \\emph{Fouxe-Rabinovitch} group \\(\\FR(G^{*n})\\), the subgroup of \\(\\Aut(G^{*n})\\) generated by partial conjugations, which conjugate the \\(i\\)-th free factor by the \\(j\\)-th free factor for some \\(i \\ne j\\), and leaves fixed all factors besides the \\(i\\)-th.\n\nDefine the \\emph{symmetric automorphism group} \\(\\Sigma\\Aut(G^{*n})\\) by\n\\[\\Sigma\\Aut(G^{*n}) = \\FR(G^{*n}) \\rtimes (\\Aut(G)\\,\\wr\\,S_n)\\]\nWhen \\(G\\) is freely indecomposable and not isomorphic to \\(\\mathbb{Z}\\), then \\(\\Sigma\\Aut(G^{*n})\\) is in fact all of \\(\\Aut(G^{*n})\\).\n\nThese groups have interesting geometric interpretations. For example, if \\(G = \\mathbb{Z}\\) then \\(\\Sigma\\Aut(G^{*n})\\) is the fundamental group of \\(n\\) unknotted circles embedded in \\(\\mathbb{R}^3\\), or the \\emph{group of string motions}, and \\(\\FR(G^{*n})\\) is the corresponding group of \\emph{pure} string motions. More generally, Griffin \\cite{Gr} constructed a classifying space for \\(\\FR(G^{*n})\\), which he calls the \\emph{moduli space of cactus products}.\n\nThe cohomology of these groups has also been much studied. For example, Hatcher-Wahl \\cite{HW} and Collinet-Djament-Griffin \\cite{CDG} proved that \\(\\Sigma\\Aut(G^{*n})\\) satisfy homological stability. On the other hand, the cohomology of \\(\\FR(G^{*n})\\) does not stabilize: e.g., \\(H^1(\\FR(G^{*n})) = H^1(G)^{\\oplus n(n-1)}\\). \n\nHowever, notice that the exact sequence\n\\[1 \\to \\FR(G^{*n}) \\to \\Sigma\\Aut(G^{*n}) \\to \\Aut(G) \\wr S_n \\to 1\\]\ngives an action of \\(\\Aut(G) \\wr S_n\\) on \\(H^i(\\FR(G^{*n}); \\mathbb{Q})\\). In fact, inner automorphisms of \\(G\\) act trivially on cohomology, so in fact we get a representation of \\(\\Out(G) \\wr S_n\\) on \\(H^i(\\FR(G^{*n}); \\mathbb{Q})\\). Wilson \\cite[Thm 7.3]{Wi} proved that \\(H^*(\\FR(\\mathbb{Z}^{*n}))\\) is a finite type \\(\\FI_{\\mathbb{Z}\/2}\\)-algebra, since \\(\\Out(\\mathbb{Z}) = \\mathbb{Z}\/2\\).\n\\begin{thm}[\\bfseries Finite generation for cohomology of Fouxe-Rabinovitch groups]\nSuppose that \\(G\\) is any group such that \\(H^*(G; k)\\) is a finite type \\(k\\)-module. Then \\(H^*(\\FR(G^{*n}); k)\\) is a finite type \\(\\FI_{\\Out G}\\sharp\\)-algebra.\n\\end{thm}\nNotice that, for example, there are many groups with even no nontrivial outer automorphisms, for which Theorem 6.1 tells us that \\(H^*(\\FR(G^{*n}); \\mathbb{Q})\\) is a finite type \\(\\FI\\sharp\\)-algebra.\n\n\\begin{proof}\nWe first need to show that \\(H^*(\\FR(G^{*n}))\\) is, in the first place, an \\(\\FI_{\\Out(G)}\\sharp\\)-algebra. Griffin \\cite[Thm B]{Gr} proved that \\(H^*(\\FR(G^{*n})) \\cong H^*((G^{*n})^{n-1})\\) as an algebra, so \\(H^*(\\FR(G^{*n}))\\) is generated by \\(\\widetilde{H}^*(G)^{\\oplus n(n-1)} \\cong \\bigoplus_{i \\ne j} \\widetilde{H}^*(G)\\). Under this isomorphism, for any \\(\\alpha \\in \\widetilde{H}^*(G)\\), write \\(\\alpha_{i,j}\\) for the element \\(\\alpha_{i,j} \\in \\bigoplus_{i \\ne j} \\widetilde{H}^*(G)\\) sitting in the \\((i,j)\\)-th summand. The map induced by an \\(\\FI_{\\Out(G)}\\sharp\\) morphism \\((s,\\vec \\phi): [m] \\to [n]\\) is:\n\\begin{align*} (s,\\vec \\phi)_*: \\bigoplus_{i \\ne j} \\widetilde{H}^*(G) &\\to \\bigoplus_{i \\ne j} \\widetilde{H}^*(G) \\\\\n\\alpha_{i,j} & \\mapsto \\begin{cases} \\phi^*_j\\,\\alpha_{s(i), s(j)} & \\text{if } i, j \\in \\operatorname{domain}(s) \\\\\n0 & \\text{otherwise} \\end{cases}\n\\end{align*}\nTherefore \\(V_n = \\bigoplus_{i \\ne j} \\widetilde{H}^*(G)_{i,j}\\) has the structure of an \\(\\FI_{\\Out(G)}\\sharp\\)-module. \\(V\\) is generated in finite degree, since it is just generated by \\(\\widetilde{H}^*(G)_{1,2}\\), and each \\(V^i_n\\) is finite-dimensional, since by assumption each \\(H^j(G)\\) is, so therefore \\(V\\) is finitely generated. Furthermore, the action of \\(\\FI_{\\Out(G)}\\sharp\\) on \\(V\\) extends to an algebra map of \\(H^*(\\FR(G^{*n}))\\), which is thus an \\(\\FI_{\\Out(G)}\\sharp\\)-algebra. Since it is generated by the finite type module \\(V\\), then \\(H^*(\\FR(G^{*n}))\\) is finite type by \\thref{alg}.\n\\end{proof}\n\\begin{cor}[\\bfseries Representation stability for cohomology of Fouxe-Rabinovitch groups]\nSuppose that \\(G\\) is any group with \\(H^*(G; \\mathbb{Q})\\) of finite type and \\(\\Out G\\) finite. Then for each \\(i\\), the characters of the \\(\\Out(G) \\wr S_n\\)-representations \\(H^i(\\FR(G^{*n}); \\mathbb{C})\\) are given by a single character polynomial of degree \\(2i\\) for all \\(n\\), and therefore \\(H^i(\\FR(G^{*n}); \\mathbb{C})\\) is representation stable, with stability degree \\(4i\\).\n\\end{cor}\n\\begin{proof}\nAs shown in the proof of Theorem 7.1, \\(H^*(\\FR(G^{*n})\\) is generated as an algebra by \\(V\\), which is generated as an \\(\\FI_G\\)-module by \\(\\widetilde{H}^*(G)_{1,2}\\), and so \\(V\\) is generated in degree 2. By \\thref{alg}, \\(H^i\\) is generated in degree \\(2i\\), and so the stable range follows from \\thref{sharp_range}.\n\\end{proof}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}